INDE 2333 Exam Questions and Answers.

INDE 2333 Exam Questions and Answers Role of statistics in engineering problem solving process - Correct Answers: - Make decisions - Solve problems - Design products & processes The field of statistics deals with the collection, presentation, analysis, and use of data - Correct Answers: True Statistics is the science of data - Correct Answers: True How variability affects the data collected and used for engineering decision - Correct Answers: Statistical techniques are useful to describe and understand variability Statistics gives us a framework for describing this variability and for learning about potential sources of variability. Note that successive observations of a system or phenomenon do not produce exactly the same result - Correct Answers: True Statistics gives us a framework for describing this variability and for learning about potential sources of variability. - Correct Answers: True the different methods that engineers use to collect data. - Correct Answers: retrospective observational designed experiment retrospective study - Correct Answers: - study using historical data - Data collected in the past for other purposes observational study - Correct Answers: Data, presently collected, by a passive observer. designed experiment - Correct Answers: Data collected in response to process input changes. built from underlying knowledge of the basic physical mechanisms that relate several variables. The form of the function is known. - Correct Answers: mechanistic model built from engineering and scientific knowledge of the phenomenon, but is not directly developed from theoretical or first-principles understanding of the underlying mechanism. The form of the function is not known a priori. - Correct Answers: empirical model provides the framework for the study and application of statistics - Correct Answers: Probability help quantify the risks involved in statistical inference, that is, risks involved in decisions made every day. - Correct Answers: Probability models Enumerative studies - Correct Answers: Control chart of past production lots. Used for lot-by-lot acceptance sampling. Analytic studies - Correct Answers: Real-time control of a production process. To identify when the shift occurs, a control chart is used. - Correct Answers: true The lesson of the Deming experiment is that a process should not be adjusted in response to random variation, but only when a clear shift in the process value becomes apparent. - Correct Answers: true Experiment - Correct Answers: a procedure that is • carried out under controlled conditions, and • executed to discover an unknown result. Long-term average is plotted as the center-line. - Correct Answers: True Long-term usual variability is plotted as upper and lower control limits around the long-term average - Correct Answers: True Probability provides the framework for the study and application of statistics - Correct Answers: True A random experiment is an experiment that results in different..... - Correct Answers: outcomes even when repeated in the same manner every time The set of all possible outcomes of a random experiment is called the sample space, S - Correct Answers: True S is discrete if it consists of a.... - Correct Answers: finite or countable infinite set of outcomes. S is continuous if it contains.... - Correct Answers: an interval of real numbers. An event (E) is a subset of the sample space of a random experiment. - Correct Answers: True a subset of the sample space of a random experiment. - Correct Answers: Event (E) Events can be discrete or continuous - Correct Answers: True Two events consists of all outcomes that are contained in one event or the other - Correct Answers: Union of two events E1 U E2 Two events consists of all outcomes that are contained in one event and the other, - Correct Answers: The Intersection of two events E1 ∩ E2 Event is the set of outcomes in the sample space that are not contained in the event - Correct Answers: The Complement of an event E' Events A and B are mutually exclusive because... - Correct Answers: they share no common outcomes The occurrence of one event precludes the occurrence of the other. - Correct Answers: True Symbolically, A ∩ B = Ø Event order is unimportant - Correct Answers: Commutative law A ∩ B = B ∩ A and A ∪ B = B ∪ A Operations flip like in algebra - Correct Answers: Distributive law (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) Grouping doesn't matter - Correct Answers: Associative law (A ∪ B) ∪ C = A ∪ (B ∪C) (A ∩ B) ∩ C = A ∩ (B ∩ C) (A ∪ B)′= A′ ∩ B′ The complement of the union is the intersection of the complements - Correct Answers: True. DeMorgan's law (A ∩ B)′= A′ ∪ B′ The complement of the intersection is the union of the complements - Correct Answers: True DeMorgan's law State the Complement law - Correct Answers: (A′)′= A. Counting techniques are used to determine the number of .... - Correct Answers: Outcomes in events Name the three counting rules - Correct Answers: Multiplication rule, Permutation rule, Combination rule The total number of ways to perform k steps is: n1 · n2 · ... · nk - Correct Answers: Multiplication rule Number of permutations for a set of n items is n! - Correct Answers: Permutation rule Order does not matter with combinations. Divide the # of permutations by r!, where r! is the # of arrangements of r elements. - Correct Answers: Combination rule C = (n r) = n!/r!(n-r)! Probability is the likelihood or chance that a particular outcome or event from a random experiment will occur - Correct Answers: True Probability is a number in the - Correct Answers: [0,1] interval. A probability of: 1 means... - Correct Answers: certainty (ALWAYS TRUE) A probability of: 0 means... - Correct Answers: impossibility (NEVER TRUE) When a sample space consists of N possible outcomes that are equally likely, then the probability of each outcome is... - Correct Answers: 1/N For a discrete sample space: P(E), the probability of an event E, equals.... - Correct Answers: The sum of the probabilities of ALL of the outcomes in E. Joint events are generated by applying basic set operations to individual events - Correct Answers: True Name the Joint events generated when a basic set of operations are applied to them... - Correct Answers: • Unions of events, A ∪ B P(A U B) = P(A) + P(B) - P(A n B) If A & B are mutually exclusive, then P(A∪ B) = P(A) + P(B) • Intersections of events, A ∩ B • Complements of events, A′ P(B |A) is the probability of event B occurring, given that... - Correct Answers: Event A has already occurred. P(B | A) = P(A∩B) / P(A) for P(A) > 0 Re-written: P(A∩B) = P(B|A)·P(A) = P(A|B)·P(B) Two events are independent if any one of the following equivalent statements is true... - Correct Answers: P(A | B) = P(A) P(B | A) = P(B) P(A∩B) = P(A)·P(B) This means that occurrence of one event has no impact on the probability of occurrence of the other event Multiple events are independent IFF - Correct Answers: P(Ei1∩Ei2 ∩ ... , ∩Eik) = P(Ei1)·P(Ei2)·...·P(Eik) *P(A|B)* = *P(B|A) X P(A)*/*P(B)* for P(B)>0 - Correct Answers: Bayes' theorem Bayes Theorem with total probability for mutually exclusive and exhaustive events Ek, with B any event and P(B) > 0. - Correct Answers: P[Ei|A] = P[A|Ei] · P[Ei]/P[A|E1] · P[E1] + P[A|E2] · P[E2] + · · · + P[A|En] · P[En] . A variable that associates a number with the outcome of a random experiment is called a... - Correct Answers: random variable A random variable is denoted by an____________such as________. After the experiment is conducted, the ___________ of the random variable is denoted by a ____________ . - Correct Answers: Uppercase letter, X, measured value, lowercase letter such as x A random variable with a finite or countably infinite range. Its values are obtained by counting - Correct Answers: Discrete Random Variable A random variable with an interval (either finite or infinite) of real numbers for its range. Its values are obtained by measuring - Correct Answers: Continuous Random Variable A pmf gives the probability that a _______ random variable is __________ to some value. - Correct Answers: discrete, exactly equal P(X=x) = f(x) > 0 if.... - Correct Answers: x in the support S f(x) is a function and can be presented... - Correct Answers: Table, Graph, Formula the sum of all probabilities = - Correct Answers: 1 How do you get f(x) knowing the sum of probabilities? - Correct Answers: Just add up the x values A cdf gives the probability that a random variable is less than or equal to some value - Correct Answers: True For discrete variables: P(X > x) = F(x) > 0 if x in the support S. - Correct Answers: False. For discrete variables: P(X ≤ x) = F(x) > 0 if x in the support S. ...of a discrete random variable X is a weighted average of the possible values of X with weights f(x) equal to the probabilities. - Correct Answers: The mean [µ or E(x)] ...uses weight f(x) as the multiplier of each possible squared deviation (x − µ)^2 - Correct Answers: The variance of X [@2 or V(X)] of discrete random variable A symmetric probability distribution whereby a finite number of values are equally likely to be observed; every one of n values has equal probability f(xi) = 1/n - Correct Answers: Discrete Uniform Distribution Let X be a discrete random variable ranging from a,a+1,a+2,...,b, for a ≤ b. There are b - (a-1) values in the inclusive interval. Therefore f(x)... - Correct Answers: f(x) = 1/(b-a+1) Mean and Variance of Discrete Uniform Distribution - Correct Answers: µ = E(x) = (b+a)/2 σ2 = V(x) = [(b-a+1)2-1]/12 In Discrete Uniform Distribution the mean is the midpoint of a&b. - Correct Answers: True Let the random variable X denote the number of the 48 voice lines that are in use at a particular time. Assume that X is a discrete uniform random variable with a range of 0 to 48. Find E(X) & σ - Correct Answers: By applying the formulas of discrete uniform distribution of E(x) & σ: E(X)= (48+0)/2=24 σ= SQRT((48-0+1)^2-1/(12))=SQRT(2400/12)=14.14 With parameters n and p is the discrete probability distribution of: • The number of successes • In a sequence of n independent yes/no experiments, • Each of which yields success with probability p - Correct Answers: Binomial Distribution Give an example of a Binomial Distribution - Correct Answers: coin toss = heads or tails The random variable X that equals the number of trials that result in a success is a ______________ with parameters 0 < p < 1 and n = 1, 2, - Correct Answers: Binomial Distribution State the probability mass function of a binomial distribution... - Correct Answers: f(x) = (n)*p^x(1-p)^(n-x) (x) Binomial Coefficient Calculation: (10) ( 3 ) - Correct Answers: (10!)/(3!7!) = (10*9*8*7!)/(3*2*1*7!) = 120 Each sample of water has a 10% chance of containing a particular organic pollutant: • Assume that the samples are independent with regard to the presence of the pollutant. • Find the probability that, in the next 18 samples, exactly 2 contain the pollutant - Correct Answers: Let X = number of samples that contain the pollutant in the next 18 samples analyzed. Then X is a binomial random variable with p = 0.1 and n = 18 f(x) = (n)*p^x(1-p)^(n-x) (x) P(X=2) = (18)*(0.1)^2(0.9)^16 = 0.2835 ( 2 ) µ = E(X) = np σ2 = V(X) = np(1-p) .....? - Correct Answers: Binomial Mean and Variance Binomial distribution has: • Fixed number of trials. • Random number of successes - Correct Answers: True ____________ is the reverse of the Binomial Distribution. RANDOM number of trials. Fixed number of successes - Correct Answers: Geometric distribution _____________ independent trials with constant probability p of a success - Correct Answers: Bernoulli trials Probability density function of the Geometric distribution is - Correct Answers: f(x) = p(1-p)^(x-1) x = 1, 2, ..., the number of failures until the 1st success. 0 < p < 1, the probability of success. The geometric distribution is the discrete probability distributions... - Correct Answers: of the number X of Bernoulli trials needed to get one *success* supported on the set { 1, 2, 3, ...}... The probability that a wafer contains a large particle of contamination is 0.01. • Assume that the wafers are independent. • What is the probability that exactly 125 wafers need to be analyzed before a large particle is detected? - Correct Answers: Geometric pmf = (1 - p)k-1 * p Let X denote the number of samples analyzed until a large particle is detected. • Then X is a geometric random variable with parameter p = 0.01. P(X=125) = (0.99)124(0.01) = 0.00288 Geometric Mean & Variance - Correct Answers: u = E(X) = 1/p σ^2 = V(X) = (1-p)/p^2 The probability that a bit transmitted through a digital transmission channel is received in error is 0.1. • Assume that the transmissions are independent events • Let the random variable X denote the number of bits transmitted until the first error. • Find the mean and standard deviation. - Correct Answers: Notice the phrase random variable X denote the number of bits transmitted *until* the first error. This means we have to employ the mean and standard deviation of the geometric distribution... Mean = μ = E(X) = 1 / p = 1 / 0.1 = 10 Variance = σ2 = V(X) = (1-p) / p2 = 0.9 / 0.01 = 90 Standard deviation = √90 = 900.5 = 9.49 For a geometric random variable, the trials are__________. Thus the count of the number of trials until the next success can be started at any trial __________ the probability distribution of the random variable - Correct Answers: Independent, Without changing The implication of using a geometric model is that the system presumably___________ - Correct Answers: will not wear out For all transmissions the probability of an error remains ________. Hence the geometric distribution is said to _____________ - Correct Answers: Constant, lack any memory Mean & Variance of Negative Binomial - Correct Answers: If X is a negative binomial random variable with parameters p and r... u = E(X) =r/p σ^2 = V(X) = r(1-p) / (p^2) In a series of independent trials with constant probability of success p - Correct Answers: Negative Binomial Distribution In Negative Binomial Distribution , the random variable X which ___________________ until r successes occur is a negative binomial random variable - Correct Answers: Equals the number of trials Probability Mass Fuction of the Negative Binomial Distribution... - Correct Answers: f(x) = ( x-1 ) p^r*(1-p)^(x-r) x = r, r+1, r+2 ( r-1) The probability that a camera passes a particular test is 0.8 • The cameras perform independently. • What is the probability that the 3rd failure is obtained in 5 or fewer tests? - Correct Answers: Distribution assigned: Negative Binomial Distribution Let X denote the number of cameras tested until 3 failures. X has a negative binomial distribution with failure p = 0.2 and r = 3. f(x) = ( x-1 ) p^r*(1-p)^(x-r) x = r, r+1, r+2 ( r-1) = ( x-1) *(0.2)^3*(0.8)^(x-3) ( r-1) = 0.2^3+(3)*(0.2)^3*(0.8)+(4)*(0.2)^3*(0.8)^2 (2) (2) = 0.056 ...is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes, wherein each draw is either a success or a failure. - Correct Answers: Hypergeometric distribution In Hypergeometric distribution, A set of N objects contains: - Correct Answers: K objects classified as successes N - K objects classified as failures In Hypergeometric distribution, A _____ of size n objects is selected at random _______________ from the N objects, where K ≤ N and n ≤ N. - Correct Answers: sample, without replacement (this is the key word in hypergeometric) State the PMF of the Hypergeometric Distribution - Correct Answers: In Hypergeometric Distribution, Let the random variable X denote the number of.... - Correct Answers: successes in the sample A batch of parts contains 100 parts from supplier A and 200 parts from Supplier B. If 4 parts are selected randomly, without replacement, what is the probability that they are all from Supplier A? - Correct Answers: A batch of parts contains 100 parts from supplier A and 200 parts from Supplier B. If 4 parts are selected randomly, without replacement, What is the probability that two or more parts are from supplier A? - Correct Answers: A batch of parts contains 100 parts from supplier A and 200 parts from Supplier B. If 4 parts are selected randomly, without replacement, What is the probability that at least one part in the sample is from Supplier A? - Correct Answers: σ2 approaches the binomial variance as n /N becomes small. - Correct Answers: true State the mean and variance of the hypergeometric distribution - Correct Answers: The Poisson Distribution is a discrete probability distribution - Correct Answers: True The Poisson Distribution that expresses the probability of a - Correct Answers: given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate independent of the time since the last event. The Poisson Distribution is that expresses the ___________ of a given number of events occurring in a fixed___________ of time and/or space if these events occur with a known __________________ and are ______________of the time since the last event. - Correct Answers: probability, interval, average rate, independent State the probability density function of the Poisson Distribution... - Correct Answers: The average event rate is 2.5 goals per match: h = 2.5 - Correct Answers: Suppose that the number of flaws in a thin copper wire follows a Poisson distribution • The mean is 2.3 flaws per mm. • Find the probability of exactly 2 flaws in 1 mm of wire. - Correct Answers: Poisson Mean & Variance - Correct Answers: The mean and variance of the Poisson model are the same. - Correct Answers: True If the variance of a data is much greater than the mean, then the Poisson distribution would not be a good model for the distribution of the random variable. - Correct Answers: True One which takes an infinite number of possible values. - Correct Answers: Continuous Random Variables Continuous random variables are usually measurements. - Correct Answers: true Continuous Random Variables Examples - Correct Answers: 1. Let Y be the height of a person (a real number). 2. Let X be the volume of juice in a can. 3. Let Y be the waiting time until the next person arrives at the server A continuous random variable is not defined at specific values. Instead, it is defined over an... - Correct Answers: interval of values Continuous Random Variable is represented by the area under a curve (in advanced mathematics, this is known as an integral). - Correct Answers: true (Continuous Random Variables)The probability of observing any single value is equal to __, since the number of values which may be assumed by the random variable is infinite. - Correct Answers: 0 In the Density Curve the total area under the area is equal to 1 - Correct Answers: True In a Density Curve X is set to be the continuous random variable that may take all... - Correct Answers: values over an interval of real numbers. In a Density Curve we P(A) be the probability that X is in the set of outcomes A. P(A) = - Correct Answers: area above A and under a curve called p(x) What is p(x) in the Density Curve? - Correct Answers: Then p(x) is a density curve IFF • The curve has no negative values (p(x) > 0 for all x) • The total area under the curve is equal to 1. In Probability Density Function F(x)>0 means that... - Correct Answers: the function is always non-negative For a continuous random variable X, a probability density function is a function such that... - Correct Answers: Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes(mA). Assume that the range of X is 4.9 ≤ x ≤ 5.1 and the PDF f(x) = 5. What is the probability that a current is less than 5mA? - Correct Answers: What is the cumulative distribution function of a continuous random variable? - Correct Answers: The probability density function (PDF) f(x) is the derivative... - Correct Answers: of the cumulative distribution function (CDF) F(x). The cumulative distribution function (CDF) F(x) is the integral of the... - Correct Answers: probability density function (PDF) f(x). The time until a chemical reaction is complete (in milliseconds, ms) is approximated by this cumulative distribution function CDF: - Correct Answers: State the mean and variance of the continuous random variable... - Correct Answers: For the copper wire current measurement, the PDF is f(x) = 0.05 for 0 ≤ x ≤ 20. Find the mean and variance... - Correct Answers: If is a continuous random variable with a probability density function f(x),... - Correct Answers: Let X be the current measured in mA. • PDF f(x) = 0.05 for 0 ≤ x ≤ 20. • What is the expected value of power when the resistance is 100 ohms? Note: • Power in watts P = 10^−6*R*I^2, where I is the current in milliamperes and R is the resistance in ohms - Correct Answers: This is the simplest distribution and analogous to its discrete counterpar - Correct Answers: Continuous Uniform Distribution Probability Density Function of Continuous Uniform Distribution - Correct Answers: f(x) = 1 / (b-a) for a ≤ x ≤ b State the mean and variance of Continuous Uniform Distribution - Correct Answers: • The random variable X has a continuous uniform distribution on [4.9, 5.1]. The probability density function of X is • f(x) = 5, 4.9 ≤ x ≤ 5.1. • What is the probability that a measurement of current is between 4.95 & 5.0 mA? - Correct Answers: •Let a = 4.9 and b = 5.1 Cumulative distribution function CDF of Uniform distribution - Correct Answers: It is important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known - Correct Answers: Normal Distribution • The normal (or Gaussian) distribution is a very common continuous probability distribution - Correct Answers: True The normal distribution is useful because of the central limit theorem. - Correct Answers: True The normal distribution is sometimes informally called the ___________, even though other distributions have a bell shape. - Correct Answers: Bell Curve State the PDF of the Normal Distribution - Correct Answers: State the mean and variance of the Normal Distribution - Correct Answers: In the PDF graph of a Normal Distribution the red curve is the standard normal distribution - Correct Answers: True A standard normal random variable Z = - Correct Answers: The cumulative distribution function CDF of a standard normal random variable Z is - Correct Answers: Z is standard normal random variable.. - Correct Answers: Standardizing a Normal Random Variable - Correct Answers: Suppose that the current measurements X in a strip of wire are assumed to follow a normal distribution with µ = 10 and σ = 2 mA. • What is the probability that the current measurement is between 9 and 11 mA? - Correct Answers: The binomial and Poisson distributions become more bell-shaped and symmetric as their mean value increase - Correct Answers: True For manual calculations, the normal approximation is practical why?... - Correct Answers: Exact probabilities of the binomial and Poisson, with large means, require computation. Normal Approximations: to Binomial - Correct Answers: Normal Approximations: to Poisson - Correct Answers: The Poisson distribution approaches the Normal distribution as the mean increases - Correct Answers: True The normal distribution is a good approximation for: Binomial if ... - Correct Answers: np > 5 and n(1-p) > 5. The normal distribution is a good approximation for: Poisson if... - Correct Answers: λ > 5. A Poisson process consists of randomly located points located on some underlying mathematical space. - Correct Answers: True A completely random process since each point is statistically independent to all the other points in the process. - Correct Answers: Poisson (point) process The points in a Poisson process follow the Poisson distribution - Correct Answers: True The random variable X that equals the distance between successive events of a Poisson process: - Correct Answers: is an exponential random variable with parameter λ • where λ> 0 = the mean number of events λ per unit interval. State the PDF of the Exponential Distribution.. - Correct Answers: Exponential distribution - Mean & Variance - Correct Answers: Poisson distribution : Mean and variance are same. - Correct Answers: True Exponential distribution : Mean and standard deviation are same. - Correct Answers: True In a large corporate computer network, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. • What is the probability that there are no log-ons in the next 6 minutes (0.1 hour)? - Correct Answers: In a large corporate computer network, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. what is the probability that the time until the next log-on is between 2 and 3 minutes (0.033 & 0.05 hours)? - Correct Answers: In a large corporate computer network, user log-ons to the system can be modeled as a Poisson process with a mean of 25 log-ons per hour. - What is the interval of time such that the probability that no log-on occurs during the interval is 0.90? What is the mean and Variance? - Correct Answers: Exponential random variables lack memory. - Correct Answers: True The exponential distribution is the only continuous distribution with this property - Correct Answers: True Lack of Property Let X denote the time between detections of a particle with a Geiger counter. • Assume X has an exponential distribution with E(X) = 1.4 minutes. • What is the probability that a particle is detected in the next 30 seconds? - Correct Answers: A continuous random variable is not defined at specific values. - Correct Answers: True This variable is defined over an interval of values, and It is represented by the area under a curve - Correct Answers: Continuous Random Variables The probability of observing any single value is equal to 0, since the number of values which may be assumed by the random variable is infinite (Continuous Random Variables) - Correct Answers: True A family of continuous probability distributions with support x ∈(0,∞) and 2 parameters: - Correct Answers: Erlang distribution In Erlang distribution, K and h is the... - Correct Answers: a positive integer 'shape' k • a positive real 'rate' h History of the Erlang Distribution - Correct Answers: The Erlang distribution was developed by A. K. Erlang (~1909) to examine the number of telephone calls which might be made at the same time to the operators of the switching stations A family of continuous probability distributions with two positive real parameters - Correct Answers: Gamma Distribution Define the parameters of the Gamma Distribution - Correct Answers: - A shape parameter k and a scale parameter θ (Used to model waiting times) - A shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter (Used in Bayesian statistics) - A shape parameter k and a mean parameter µ = k/β. A gamma distribution with integer k is an Erlang distribution - Correct Answers: True The Erlang distribution with k = 1 simplifies to the ______ distribution. - Correct Answers: Exponential The Erlang distribution is a generalization of the exponential distribution. - Correct Answers: True The exponential distribution models the interval to the 1st event, while the Erlang distribution models the interval to the... - Correct Answers: rth event i.e. sum of exponentials If r is not required to be an integer, then the distribution is called a Gamma distribution. - Correct Answers: True The exponential, Erlang, and Gamma distributions are based on the Poisson process. - Correct Answers: True The failures of CPUs of large computer systems are often modeled as a Poisson process. • Assume that units that fail are repaired immediately and the mean number of failures per hour is 0.0001. • Let X denote the time until 4 failures occur. • What is the probability that X exceeds 40,000 hours? - Correct Answers: The gamma function is the generalization of the factorial function for r > 0, not just non negative integers - Correct Answers: True State the properties of the Gamma Function - Correct Answers: State the PDF of Gamma Random Variable - Correct Answers: Mean & Variance of the Gamma - Correct Answers: The time to prepare a micro-array slide for high-output genomics is a Poisson process with a mean of 2 hours per slide. • What is the probability that 10 slides require more than 25 hours? - Correct Answers: Let X denote the time to prepare 10 slides. Because of the assumption of a Poisson process, X has a gamma distribution with λ = ½, r = 10, and the requested probability is P(X > 25) The time to prepare a micro-array slide for high-output genomics is a Poisson process with a mean of 2 hours per slide. • What is the probability that 10 slides require more than 25 hours? What are the mean and standard deviation of the time to prepare 10 slides? - Correct Answers: • The Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. - Correct Answers: True Used for modeling the time until failure of physical systems - Correct Answers: Weibull Distribution Parameters in Weibull Distribution - Correct Answers: Parameters can handle: • Failures increasing with time • Failures decreasing with time • Failures constant over time Weibull Distribution- Some Uses - Correct Answers: - survival analysis • reliability engineering and failure analysis • electrical engineering to represent overvoltage occurring in an electrical system • industrial engineering to represent manufacturing and delivery times • extreme value theory • weather forecasting State the PDF of the Weibull Distribution - Correct Answers: Mean and Variance of the Weibull Distribution - Correct Answers: The time to failure (in hours) of a bearing in a mechanical shaft is modeled as a Weibull random variable with β = ½ and δ = 5,000 hours. • What is the mean time until failure? - Correct Answers: The time to failure (in hours) of a bearing in a mechanical shaft is modeled as a Weibull random variable with β = ½ and δ = 5,000 hours. • What is the probability that a bearing will last at least 6,000 hours? - Correct Answers: A continuous probability distribution of a random variable whose logarithm is normally distributed - Correct Answers: Lognormal Distribution Used to describe natural phenomena where relative growth rate is independent of size - Correct Answers: Lognormal Distribution Uses of Lognormal Distribution - Correct Answers: Used in reliability analysis, wireless communication, economics and finance PDF of the Lognormal Distribution - Correct Answers: Mean and Variance of the Lognormal Distribution - Correct Answers: The lifetime of a semiconductor laser has a lognormal distribution with θ = 10 and ω = 1.5 hours. • What is the probability that the lifetime exceeds 10,000 hours? - Correct Answers: The lifetime of a semiconductor laser has a lognormal distribution with θ = 10 and ω = 1.5 hours. What is the mean and variance of the lifetime? - Correct Answers: A family of continuous probability distributions defined on the interval [0, 1] with two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. - Correct Answers: Beta Distribution In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions - Correct Answers: True PDF of the Beta Random Variable - Correct Answers: Mean of the Beta Random Variable - Correct Answers: The mode is the peak of the probability density function. - Correct Answers: True Mode formula in Beta Distribution - Correct Answers: What does the Joint Probability Mass Function satisfy? - Correct Answers: random variables X & Y in a random experiment differs from the probability distribution of each individual variable. - Correct Answers: joint probability distribution The individual probability distribution of a random variable. - Correct Answers: marginal probability distribution The marginal probability distribution for X is found by summing the probabilities in each.. - Correct Answers: column whereas the marginal probability distribution for Y is found by summing the probabilities in each row If the joint probability density function of random variables X and Y is fXY(x,y), the marginal probability density functions of X and Y are... - Correct Answers: Mean & Variance of a Marginal Distribution - Correct Answers: Mean & Variance of a Marginal Distribution Example - Correct Answers: The conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value - Correct Answers: True Covariance is a measure of the relationship between two random variables. - Correct Answers: True If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the smaller values, i.e., the variables tend to show similar behavior, the covariance is... - Correct Answers: positive • When the greater values of one variable mainly correspond to the smaller values of the other, i.e., the variables tend to show opposite behavior, the covariance is... - Correct Answers: negative The normalized version of the covariance, the correlation coefficient, shows by - Correct Answers: its magnitude the strength of the linear relation Covariance Graphs - Correct Answers: Covariance Defined - Correct Answers: 2 kinds of covariance - Correct Answers: population covariance sample covariance population covariance - Correct Answers: The population covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution. sample covariance - Correct Answers: The sample covariance, which serves as an estimated value of the population parameter. • The bivariate normal distribution is the extension of a normal distribution to 2 random variables. - Correct Answers: True A powerful feature of the bivariate normal distribution is that the conditional probability distribution function for one of the variables, given a known value for the other variable, is normally distributed - Correct Answers: True ...the numeric observations of a phenomenon of interest - Correct Answers: Data The totality of all observations is a.. - Correct Answers: population A portion used for analysis is a ... - Correct Answers: Sample mean - Correct Answers: measures the center. variance - Correct Answers: The variance measures the spread. mode - Correct Answers: the value that appears most often in a set of data. median - Correct Answers: is the number separating the higher half of a data sample, a population, or a probability distribution, from the lower half outlier - Correct Answers: r is an observation point that is distant from other observations. Variance: Sample vs. Population - Correct Answers: The standard deviation is the... - Correct Answers: square root of the variance. • s is the sample standard deviation symbol • σ is the population standard deviation symbol The population variance is calculated with N, the population size - Correct Answers: True The true variance is based on data deviations from the true mean,µ - Correct Answers: True The sample variance is calculated with n-1, the sample size minus 1 - Correct Answers: True • X-bar is an estimator of µ; close but not the same. - Correct Answers: True The sample variance is calculated with the quantity n-1. • This quantity is called the - Correct Answers: Degrees of Freedom • Steps to construct a stem-and-leaf diagram: - Correct Answers: 1). Divide each number (xi) into two parts: a stem, consisting of the leading digits, and a leaf, consisting of the remaining digit. 2) List the stem values in a vertical column. 3) Record the leaf for each observation beside its stem. 4) Write the units for the stems and leaves on the display. stem and leaf example - Correct Answers: • A histogram is a graphical representation of the distribution of numerical data - Correct Answers: True It is an estimate of the probability distribution of a continuous quantitative variable. To construct a histogram - Correct Answers: - The first step is to "bin" the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. - The bins are usually specified as consecutive, nonoverlapping intervals of a variable. - The bins (intervals) must be adjacent, and are usually equal size. • Steps to construct a histogram with equal bin widths: - Correct Answers: 1). Label the bin boundaries on the horizontal scale. 2) Mark & label the vertical scale with the frequencies or relative frequencies. 3) Above each bin, draw a rectangle whose height is equal to the frequency corresponding to that bin. • Histograms must include the features: - Correct Answers: • (1) horizontal scale bin boundaries & labels with units, • (2) vertical scale measurements and labels, • (3) histogram title at top or in legend a convenient way of graphically depicting groups of numerical data through their quartiles - Correct Answers: box plot. • A box plot shows Spread, Outliers, Center and Shape (SOCS). A box plot may also have lines extending vertically from the boxes ("whiskers") indicating variability outside the upper and lower quartiles - Correct Answers: True • A box plot is also called a box-and-whisker chart. - Correct Answers: True A box plot usually displays a 5-number summary: - Correct Answers: min, q1, median, q3, and max A sample is said to be random if it is selected in such a way that every possible sample has the same probability of being selected - Correct Answers: True • If a sample is not random, then statistical methods will not work properly. - Correct Answers: True • The purpose of taking a random sample is to obtain information about the unknown population parameters. - Correct Answers: True • Most statistical procedures used in quality improvement work are based on the assumption that the population is approximately - Correct Answers: normally distributed. • Probability plots can determine an appropriate distribution. - Correct Answers: True The distribution that we are comparing the X's to should have a mean and variance that - Correct Answers: match the sample mean and variance shows the data value, or statistic, on the vertical axis with time on the horizontal axis. - Correct Answers: A time sequence plot reveals trends, cycles or other time-oriented behavior that could not be seen in the data - Correct Answers: time sequence plot a graph that displays observed data in a time sequence - Correct Answers: time sequence plot • A time sequence plot is a form of line chart - Correct Answers: true Time sequence vs control charts - Correct Answers: Similar to the control charts used in statistical process control: • Simpler to produce than control charts. • But time sequence plots do not show the control limits of the process or allow for the full range of analytic techniques. Sometimes we want to compare between more than one sample. - Correct Answers: • We can place the boxplots of the two samples side-byside. • This will allow us to compare how the medians differ between samples, as well as the first and third quartile. • It also tells us about the difference in spread between the two samples. draws conclusions from sample data based on the frequency or proportion of the data - Correct Answers: Frequentist statistics is used to update the probability for a hypothesis as more evidence or information becomes available. - Correct Answers: • Bayesian statistics Probability is based on the assumption that any given experiment can be considered as one of an infinite sequence of possible repetitions of the same experiment, each capable of producing statistically independent results. - Correct Answers: True Probability is based on the assumption that results are conditional on prior data and beliefs. - Correct Answers: True Point estimation involves the use of sample data to calculate a single value (known as a statistic) which is to serve as a - Correct Answers: "best estimate" of an unknown (fixed or random) population parameter. the distribution of averages of many trials is always normal, even if the distribution of each trial is not - Correct Answers: Central Limit Theorem an unbiased estimator of µ. - Correct Answers: sample mean a unbiased estimator of σ2. - Correct Answers: sample variance If we consider all unbiased estimators of θ, the one with the smallest variance is called the - Correct Answers: minimum variance unbiased estimator (MVUE). mean squared deviation (MSD) of an estimator measures the average of the squares of the errors or deviations, that is, the difference between the estimator and what is estimated. - Correct Answers: mean squared error (MSE) a risk function, corresponding to the expected value of the squared error loss or quadratic loss. - Correct Answers: mean squared error (MSE) An estimator whose MSE is smaller than that of any other estimator is called an - Correct Answers: optimal estimator. A biased estimator can be preferred than an unbiased estimator if it has a - Correct Answers: smaller MSE. The MSE is an important criterion for comparing two estimators - Correct Answers: True