Chapter 2: Probability Variance: Expected value & variance & standard deviation of CRVS:
Sample spaces (S): 2
𝑛
2 Expected value & variance & standard deviation of CRVS:
- Set of all possible outcomes in a random experiment. σ = 𝑉(𝑥) = ∑ (𝑥 − µ) 𝑝(𝑥𝑖) - Expected value (population mean):
𝑖=0
- Discrete – Consists of finite or countable infinite outcomes. ∞
Standard deviation: µ = 𝐸(𝑥) = ∫ 𝑥 * 𝑓(𝑥)𝑑𝑥
- Continuous – Contains an interval of real numbers.
2 −∞
σ = σ = 𝑣(𝑥)
- Variance:
Event (E): ∞
2 2
- Subset of the sample space of a random experiment. Types of Discrete Random Variables (DRV) probability distributions: σ = 𝑉(𝑥) = ∫ (𝑥 − µ) * 𝑓(𝑥)𝑑𝑥
- Union ( E1 ∪ E2) ; Discrete uniform distribution: −∞
- Intersection (E1 ∩ E2) ; - DRV X assumes the values a, a + 1,...,B − 1,b (consecutive integers) with - Standard deviation:
- Complement (𝐸' 𝑂𝑅 𝐸 )
𝐶 equal probabilities. 2
σ = σ = 𝑣(𝑥)
1
PMF: P(X) = 𝑏−𝑎+1
Probability of an event: Mean: µ = 𝐸(𝑋) =
𝑎+𝑏 Continuous uniform distribution:
2
- P(E) – Likelihood of event “E” occurring. 2
- CRV X assumes values in (a, b) with equal probabilities:
2 (𝑏−𝑎+1) −1
- Lies between [0,1] (0 is impossible to occur, 1 is with a certain to occur) Variance: σ = 𝑉(𝑋) = 12 PMF: P(X) =
1
𝑏−𝑎
Combination Rule: 𝑎+𝑏
- Selection of r items from a set of n items, where order does not matter. Mean: µ = 𝐸(𝑋) =
BERNOULLI TRIAL: 2
𝑛! 2
2
𝑛𝑐𝑟 = 𝑟!(𝑛−𝑟)! - DRV X assumes one of two outcomes (success = 1, failure = 0) Variance: σ = 𝑉(𝑋) =
(𝑏−𝑎)
12
- Success has probability p and failure has probability 1− p
𝑥 1−𝑥
Addition rule: PMF: 𝑝(𝑥) = 𝑝 (1 − 𝑝)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) Mean: µ = 𝐸(𝑋) = 𝑝
2
Variance: σ = 𝑉(𝑋) = 𝑝(1 − 𝑝)
Two events A and B are:
- Mutually exclusive if P(A ∩ B) = 0 (Cannot occur at the same time)
Binomial distribution
- Independent then P(A ∩ B) = P(A) ⋅ P(B)(Don’t influence each other’s
- DRV X is the number of successes in n bernoulli trials
likelihood) 𝑥 𝑛−𝑥
PMF: 𝑝(𝑥) = (𝑛𝐶𝑥)𝑝 (1 − 𝑝) (𝑓𝑜𝑟 𝑥 = 1, 2,...)
Conditional probability: Mean: µ = 𝐸(𝑋) = 𝑛𝑝
2
P(A | B) : the probability that event A has occurred given that event B has Variance: σ = 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝) Normal distribution:
already occurred 2
X ∼ n(μ, σ ) is used to denote that CRV X is drawn from the normal
P(A ∩ B) = P(A)P(B | A) = P(B)P(A | B) Poisson distribution 2
-DRV X is the number of events in a poisson process (where events occur distribution. μ is the mean and σ is the variance.
Total probability rule: randomly but at a constant rate)
S: Sample space, E1, E2, . . . , En : Partition of S 𝑒
−λ𝑇 𝑥
(λ𝑇)
Normalcdf( L, U, μ, σ) : finds the probability between two values L and U on
PMF: 𝑝(𝑥) = 𝑓𝑜𝑟 𝑥 = 1, 2, 3,... a calculator. No need to integrate the pdf
Then 𝑥!
P(B) = P(B | E1)P(E1) + P(B | E2)P(E2) + . . . + P(B | En)P(En) Mean: µ = 𝐸(𝑋) = λ𝑡
2
Variance: σ = 𝑉(𝑋) = λ𝑡 Standard normal distribution:
2
Bayes’ theorem: x- quantity, T- scale factor, λ -rate Normal distribution with μ = 0 and σ = 1
P(A | B) = P(B | A) × P(A) ( when P(B) > 0 ) The standard normal random variable is denoted as Z, and for any x ∼ n(μ,
2
Hypergeometric distribution σ ), we can find Z with the following equation:
Random variables: A set of N objects contain: 𝑋−µ
𝑍=
A function that assigns a real number to each outcome in the sample space - D objects as successes σ
of a - N-D objects as failures
random experiment. - A sample size n is selected randomly
X – Random variable (𝐷𝐶𝑥)((𝑁−𝐷)𝐶(𝑛−𝑥))
𝑝(𝑥) = (𝑁𝐶𝑛)
x– measured value of that particular random variable
𝑛𝐷
µ= 𝑁
CHAPTER 3: DISCRETE RANDOM VARIABLES & PROBABILITY 2 𝑛𝐷 𝐷 𝑁−𝑛
σ = (1 − )( 𝑁−1 )
DISTRIBUTIONS 𝑁 𝑁
Probability Mass Function (PMF) CHAPTER 4: CONTINUOUS RANDOM
- p(x) : the probability function of a discrete random variable VARIABLES & PROBABILITY DISTRIBUTIONS
Probability density function (PDF) Z-score: value on the Z-axis where the area to the right of the value is equal
- Three properties to know:
- f(x) : the probability function of a continuous random variable. to the given probability.
1. 𝑝(𝑋𝑖) ≥ 0
- Three properties: Find area given Z-score: normalcdf(-∞,zscore, μ = 0,σ = 1)
𝑛
1. f(x) ≥ 0 Find Z-score given area: invNorm(area, μ = 0,σ = 1,Tail = Left))
2. ∑ 𝑝(𝑥𝑖) = 1
∞
𝑖=1
3. 𝑝(𝑥𝑖) = 𝑝(𝑋 = 𝑥𝑖)(the definition of pmf) 2. ∫ 𝑓(𝑥)𝑑𝑥 = 1
−∞
𝑎
Cumulative distribution function (CDF): 3. P(a ≤ X ≤ b) =∫ 𝑓(𝑥)𝑑𝑥
𝑏
- F(x): probability that a random variable X is less than or equal to x
- Three properties to know
Cumulative distribution function (cdf) for continuous random
1. F(x) = P(X ≤ x)
variables:
2. 0 ≤ F(x) ≤ 1
- F(x): probability that a random variable X is less than or equal to x.
3. If x ≤ y, then F(x) ≤ F(y)
- Two properties to know: EXPONENTIAL DISTRIBUTION:
Expected value & variance & standard deviation:
1. F(x) = P(X ≤ x) = P(X < x ) - CRV X assumes a distance between successive events in a poisson
Expected value (population mean): 𝑥
𝑛
2. F(x) = ∫ 𝑓(𝑥)𝑑𝑥 process with mean number of events λ > 0 per unit interval.
µ = 𝐸(𝑋) = ∑ 𝑝(𝑥𝑖) * 𝑥𝑖 −∞
𝑖=0
Notice that P(x1≤ X ≤ x2) = P(X ≤ x2) − P(X ≤ x1) = F(x2) − F(x1) - Pdf: 𝑓(𝑥) = λ𝑒
−λ𝑥
𝑓𝑜𝑟 𝑥 ≥ 0
, −λ𝑥
- Cdf: 𝐹(𝑥) = 1 − λ𝑒 𝑓𝑜𝑟 𝑥 ≥ 0 Binomial Distribution: 2nd -> vars -> binomalcdf (odd of x successes) OR
- Mean :µ = 𝐸(𝑥) = 1/λ binomalcdf (odd of equal or less than x successes) Quiz 2:
2 2 1. The notation, F(x), has the same meaning as P(X < x): True
- Variance: σ = 𝑉(𝑋) = 1/λ GIGA CHAD calculator keys(CASIO fx-115ES PLUS): 2. For a function to be considered a probability mass function (pmf), the
Set to mode 1:COMP following must be true: False
Chapter 6:The Role of Statistics in Engineering: Permutations: SHIFT + *
Numerical Summaries of Data Combinations: SHIFT + /
• Data are the numeric observations of a phenomenon of interest.
• The totality of all observations is a population. Formulas:
• A portion used for analysis is a sample. Continuous:
• The center is measured by the mean. - Uniform distribution:
• The spread is measured by the standard deviation. 1
PMF: P(X) = 𝑏−𝑎
𝑎+𝑏
Sample Mean: Mean: µ = 𝐸(𝑋) = 2 3. Consider the following probability mass function (pmf):
- For the 𝑁 observations in a population denoted by x1, x2, …, xN: 2 (𝑏−𝑎)
2
p(x) = 0.25 for x = 0, 1, 2, 3
𝑁 𝑥 Variance: σ = 𝑉(𝑋) =
𝑖
12 Random variable, X, is Discrete
µ= ∑
𝑖=1
𝑁
4. The notation, p(x), has the same meaning as: P (X=x)
- Normal distribution:
- For the n observations in a random samplede noted by x1, x2, …, xn: 1 𝑥−µ 2
−2( )
𝑛 𝑥𝑖 PDF: 𝑓(𝑥) =
𝑒 σ
Quiz 3:
𝑥= ∑ 𝑛
σ 2π 1. If a random variable, X, measuring the number of customers that enter a
𝑖=1
bank is assumed to be Normally distributed, then the distribution of
- Exponential distribution: inter-arrival times to that bank is exponential: False
Population and Sample Standard Deviation −λ𝑥
2
PDF: 𝑓(𝑥) = λ𝑒 𝑓𝑜𝑟 𝑥 ≥ 0 2. You can determine the probability density function of a continuous random
The population variance is σ Mean :µ = 𝐸(𝑥) = 1/λ variable by taking the derivative of the cumulative distribution function: True
The population standard deviation is 𝝈. 2 2
2 Variance: σ = 𝑉(𝑋) = 1/λ
The sample variance is 𝑠 . Quiz 4:
The sample standard deviation is 𝒔. 1. A histogram is used primarily for what type of data? Numerical
Discrete:
2. The totality of all observations is a sample. False
- Uniform distribution:
Variance: 1
PMF: P(X) =
- For the 𝑁 observations in a population denoted by x1, x2, …, xN: 𝑏−𝑎+1 Quiz 5:
𝑎+𝑏
2
𝑁 (𝑥 −µ)2 Mean: µ = 𝐸(𝑋) = 1. There is no difference between the z distribution and the t distribution.
𝑖 2
σ = ∑ 𝑁 2 False
2 (𝑏−𝑎+1) −1
𝑖=1 Variance: σ = 𝑉(𝑋) = 2
- For the n observations in a random samplede noted by x1, x2, …, xn:
12 2. The 𝑋 distribution and statistic are used to develop confidence intervals
𝑛 (𝑥𝑖−𝑥)
2 around which of the following values? Population variance
2
𝑠 = ∑ - Binomial distribution:
𝑛−1
𝑖=1 𝑥 𝑛−𝑥
PMF: 𝑝(𝑥) = (𝑛𝐶𝑥)𝑝 (1 − 𝑝) (𝑓𝑜𝑟 𝑥 = 1, 2,...) Quiz 7:
Mean: µ = 𝐸(𝑋) = 𝑛𝑝 Two machines are used for filling plastic bottles with a net volume of 16.0
Frequency Distributions 2 ounces. The fill volumes can be assumed to be normal with standard
• A frequency distribution is a compact summary of data, Variance: σ = 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝)
deviation equal to 0.020 and 0.025 ounces. A member of the quality
expressed as a table, graph, or function. engineering staff suspects that both machines fill to the same mean net
• The data is gathered into bins, defined by class intervals. - Poisson distribution:
−λ 𝑥 volume, whether or not this volume is 16.0 ounces. A random sample of 10
• Rule of Thumb: the number of bins should be the square root of the PMF: 𝑝(𝑥) =
𝑒 (λ)
𝑓𝑜𝑟 𝑥 = 1, 2, 3,... bottles is taken from the output of each machine.
𝑥!
sample size, 𝒏.
Mean: µ = 𝐸(𝑋) = λ
2 What type of hypothesis test would be most appropriate for this problem?
Variance: σ = 𝑉(𝑋) = λ
Two Sample Z-Test
x- quantity, λ -rate
Quiz 8:
- Geometric distribution:
𝑥−1
1. A small correlation coefficient, let's say less than 0.05, means there is no
PMF: 𝑝(𝑥) = 𝑝(1 − 𝑝) relationship between the independent and dependent variables. False
Histograms
Mean: µ = 𝐸(𝑋) = 1/𝑝
A histogram is a visual display of a frequency distribution 2. The residual can be found by the following equation: 𝑒𝑖 = 𝑦𝑖 − 𝑦𝑖. False
2 2
Steps to construct a histogram with equal bin widths: Variance: σ = 𝑉(𝑋) = (1 − 𝑝)/𝑝
1) Label the bin boundaries on the horizontal scale. 3. In simple linear regression, the values β1 and β0 are estimates for the
2) Mark and label the vertical scale with the frequencies or relative - Hypergeometric distribution: slope and intercept, respectively, of the line that relates an independent
frequencies. 𝑝(𝑥) =
(𝐷𝐶𝑥)((𝑁−𝐷)𝐶(𝑛−𝑥))
variable to a dependent variable. True
(𝑁𝐶𝑛)
3) Above each bin, draw a rectangle whose height is equal to the frequency 𝑛𝐷 4. Which of the following methods is being used to generate the estimates of
µ=
corresponding to that bin. 𝑁 the coefficients for a linear regression equation? Ordinary Least Squares
2 𝑛𝐷 𝐷 𝑁−𝑛
σ = 𝑁
(1 − 𝑁
)( 𝑁−1 )
Practice Test 1:
Quiz questions: 1. Consider the following function: f(t) = 2kt3, where 2 ≤ t ≤ 4. For what
Quiz 1: value of k does this function become a probability density function?
1. If two events are independent of one another then the probability of their a. 1/136 b. 1/116 c. 1/260 d. 1/120 e. Answer not given
intersection is zero: False
2. An experiment that results in the same outcome when replicated is
Calculator keys (TI-84): referred to as a random experiment: True
Summation: Math -> 0:summation 3. A random variable is said to be discrete if it has an infinite number of
𝑛! values in a finite interval of its range: False
Permutations: Math -> Prob -> nPr (𝑛𝑃𝑟 = )
(𝑛−𝑟)! 4. The combination rule determines the number of ways a selection of x
𝑛!
Combinations: Math -> Prob -> nCr (𝑛𝐶𝑟 = 𝑟!(𝑛−𝑟)!
) items from a set of n available items can happen if we assume order does
Normal Distribution: 2nd -> vars -> normalcdf (use 1EE99 for infinity) not matter: True
Sample spaces (S): 2
𝑛
2 Expected value & variance & standard deviation of CRVS:
- Set of all possible outcomes in a random experiment. σ = 𝑉(𝑥) = ∑ (𝑥 − µ) 𝑝(𝑥𝑖) - Expected value (population mean):
𝑖=0
- Discrete – Consists of finite or countable infinite outcomes. ∞
Standard deviation: µ = 𝐸(𝑥) = ∫ 𝑥 * 𝑓(𝑥)𝑑𝑥
- Continuous – Contains an interval of real numbers.
2 −∞
σ = σ = 𝑣(𝑥)
- Variance:
Event (E): ∞
2 2
- Subset of the sample space of a random experiment. Types of Discrete Random Variables (DRV) probability distributions: σ = 𝑉(𝑥) = ∫ (𝑥 − µ) * 𝑓(𝑥)𝑑𝑥
- Union ( E1 ∪ E2) ; Discrete uniform distribution: −∞
- Intersection (E1 ∩ E2) ; - DRV X assumes the values a, a + 1,...,B − 1,b (consecutive integers) with - Standard deviation:
- Complement (𝐸' 𝑂𝑅 𝐸 )
𝐶 equal probabilities. 2
σ = σ = 𝑣(𝑥)
1
PMF: P(X) = 𝑏−𝑎+1
Probability of an event: Mean: µ = 𝐸(𝑋) =
𝑎+𝑏 Continuous uniform distribution:
2
- P(E) – Likelihood of event “E” occurring. 2
- CRV X assumes values in (a, b) with equal probabilities:
2 (𝑏−𝑎+1) −1
- Lies between [0,1] (0 is impossible to occur, 1 is with a certain to occur) Variance: σ = 𝑉(𝑋) = 12 PMF: P(X) =
1
𝑏−𝑎
Combination Rule: 𝑎+𝑏
- Selection of r items from a set of n items, where order does not matter. Mean: µ = 𝐸(𝑋) =
BERNOULLI TRIAL: 2
𝑛! 2
2
𝑛𝑐𝑟 = 𝑟!(𝑛−𝑟)! - DRV X assumes one of two outcomes (success = 1, failure = 0) Variance: σ = 𝑉(𝑋) =
(𝑏−𝑎)
12
- Success has probability p and failure has probability 1− p
𝑥 1−𝑥
Addition rule: PMF: 𝑝(𝑥) = 𝑝 (1 − 𝑝)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) Mean: µ = 𝐸(𝑋) = 𝑝
2
Variance: σ = 𝑉(𝑋) = 𝑝(1 − 𝑝)
Two events A and B are:
- Mutually exclusive if P(A ∩ B) = 0 (Cannot occur at the same time)
Binomial distribution
- Independent then P(A ∩ B) = P(A) ⋅ P(B)(Don’t influence each other’s
- DRV X is the number of successes in n bernoulli trials
likelihood) 𝑥 𝑛−𝑥
PMF: 𝑝(𝑥) = (𝑛𝐶𝑥)𝑝 (1 − 𝑝) (𝑓𝑜𝑟 𝑥 = 1, 2,...)
Conditional probability: Mean: µ = 𝐸(𝑋) = 𝑛𝑝
2
P(A | B) : the probability that event A has occurred given that event B has Variance: σ = 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝) Normal distribution:
already occurred 2
X ∼ n(μ, σ ) is used to denote that CRV X is drawn from the normal
P(A ∩ B) = P(A)P(B | A) = P(B)P(A | B) Poisson distribution 2
-DRV X is the number of events in a poisson process (where events occur distribution. μ is the mean and σ is the variance.
Total probability rule: randomly but at a constant rate)
S: Sample space, E1, E2, . . . , En : Partition of S 𝑒
−λ𝑇 𝑥
(λ𝑇)
Normalcdf( L, U, μ, σ) : finds the probability between two values L and U on
PMF: 𝑝(𝑥) = 𝑓𝑜𝑟 𝑥 = 1, 2, 3,... a calculator. No need to integrate the pdf
Then 𝑥!
P(B) = P(B | E1)P(E1) + P(B | E2)P(E2) + . . . + P(B | En)P(En) Mean: µ = 𝐸(𝑋) = λ𝑡
2
Variance: σ = 𝑉(𝑋) = λ𝑡 Standard normal distribution:
2
Bayes’ theorem: x- quantity, T- scale factor, λ -rate Normal distribution with μ = 0 and σ = 1
P(A | B) = P(B | A) × P(A) ( when P(B) > 0 ) The standard normal random variable is denoted as Z, and for any x ∼ n(μ,
2
Hypergeometric distribution σ ), we can find Z with the following equation:
Random variables: A set of N objects contain: 𝑋−µ
𝑍=
A function that assigns a real number to each outcome in the sample space - D objects as successes σ
of a - N-D objects as failures
random experiment. - A sample size n is selected randomly
X – Random variable (𝐷𝐶𝑥)((𝑁−𝐷)𝐶(𝑛−𝑥))
𝑝(𝑥) = (𝑁𝐶𝑛)
x– measured value of that particular random variable
𝑛𝐷
µ= 𝑁
CHAPTER 3: DISCRETE RANDOM VARIABLES & PROBABILITY 2 𝑛𝐷 𝐷 𝑁−𝑛
σ = (1 − )( 𝑁−1 )
DISTRIBUTIONS 𝑁 𝑁
Probability Mass Function (PMF) CHAPTER 4: CONTINUOUS RANDOM
- p(x) : the probability function of a discrete random variable VARIABLES & PROBABILITY DISTRIBUTIONS
Probability density function (PDF) Z-score: value on the Z-axis where the area to the right of the value is equal
- Three properties to know:
- f(x) : the probability function of a continuous random variable. to the given probability.
1. 𝑝(𝑋𝑖) ≥ 0
- Three properties: Find area given Z-score: normalcdf(-∞,zscore, μ = 0,σ = 1)
𝑛
1. f(x) ≥ 0 Find Z-score given area: invNorm(area, μ = 0,σ = 1,Tail = Left))
2. ∑ 𝑝(𝑥𝑖) = 1
∞
𝑖=1
3. 𝑝(𝑥𝑖) = 𝑝(𝑋 = 𝑥𝑖)(the definition of pmf) 2. ∫ 𝑓(𝑥)𝑑𝑥 = 1
−∞
𝑎
Cumulative distribution function (CDF): 3. P(a ≤ X ≤ b) =∫ 𝑓(𝑥)𝑑𝑥
𝑏
- F(x): probability that a random variable X is less than or equal to x
- Three properties to know
Cumulative distribution function (cdf) for continuous random
1. F(x) = P(X ≤ x)
variables:
2. 0 ≤ F(x) ≤ 1
- F(x): probability that a random variable X is less than or equal to x.
3. If x ≤ y, then F(x) ≤ F(y)
- Two properties to know: EXPONENTIAL DISTRIBUTION:
Expected value & variance & standard deviation:
1. F(x) = P(X ≤ x) = P(X < x ) - CRV X assumes a distance between successive events in a poisson
Expected value (population mean): 𝑥
𝑛
2. F(x) = ∫ 𝑓(𝑥)𝑑𝑥 process with mean number of events λ > 0 per unit interval.
µ = 𝐸(𝑋) = ∑ 𝑝(𝑥𝑖) * 𝑥𝑖 −∞
𝑖=0
Notice that P(x1≤ X ≤ x2) = P(X ≤ x2) − P(X ≤ x1) = F(x2) − F(x1) - Pdf: 𝑓(𝑥) = λ𝑒
−λ𝑥
𝑓𝑜𝑟 𝑥 ≥ 0
, −λ𝑥
- Cdf: 𝐹(𝑥) = 1 − λ𝑒 𝑓𝑜𝑟 𝑥 ≥ 0 Binomial Distribution: 2nd -> vars -> binomalcdf (odd of x successes) OR
- Mean :µ = 𝐸(𝑥) = 1/λ binomalcdf (odd of equal or less than x successes) Quiz 2:
2 2 1. The notation, F(x), has the same meaning as P(X < x): True
- Variance: σ = 𝑉(𝑋) = 1/λ GIGA CHAD calculator keys(CASIO fx-115ES PLUS): 2. For a function to be considered a probability mass function (pmf), the
Set to mode 1:COMP following must be true: False
Chapter 6:The Role of Statistics in Engineering: Permutations: SHIFT + *
Numerical Summaries of Data Combinations: SHIFT + /
• Data are the numeric observations of a phenomenon of interest.
• The totality of all observations is a population. Formulas:
• A portion used for analysis is a sample. Continuous:
• The center is measured by the mean. - Uniform distribution:
• The spread is measured by the standard deviation. 1
PMF: P(X) = 𝑏−𝑎
𝑎+𝑏
Sample Mean: Mean: µ = 𝐸(𝑋) = 2 3. Consider the following probability mass function (pmf):
- For the 𝑁 observations in a population denoted by x1, x2, …, xN: 2 (𝑏−𝑎)
2
p(x) = 0.25 for x = 0, 1, 2, 3
𝑁 𝑥 Variance: σ = 𝑉(𝑋) =
𝑖
12 Random variable, X, is Discrete
µ= ∑
𝑖=1
𝑁
4. The notation, p(x), has the same meaning as: P (X=x)
- Normal distribution:
- For the n observations in a random samplede noted by x1, x2, …, xn: 1 𝑥−µ 2
−2( )
𝑛 𝑥𝑖 PDF: 𝑓(𝑥) =
𝑒 σ
Quiz 3:
𝑥= ∑ 𝑛
σ 2π 1. If a random variable, X, measuring the number of customers that enter a
𝑖=1
bank is assumed to be Normally distributed, then the distribution of
- Exponential distribution: inter-arrival times to that bank is exponential: False
Population and Sample Standard Deviation −λ𝑥
2
PDF: 𝑓(𝑥) = λ𝑒 𝑓𝑜𝑟 𝑥 ≥ 0 2. You can determine the probability density function of a continuous random
The population variance is σ Mean :µ = 𝐸(𝑥) = 1/λ variable by taking the derivative of the cumulative distribution function: True
The population standard deviation is 𝝈. 2 2
2 Variance: σ = 𝑉(𝑋) = 1/λ
The sample variance is 𝑠 . Quiz 4:
The sample standard deviation is 𝒔. 1. A histogram is used primarily for what type of data? Numerical
Discrete:
2. The totality of all observations is a sample. False
- Uniform distribution:
Variance: 1
PMF: P(X) =
- For the 𝑁 observations in a population denoted by x1, x2, …, xN: 𝑏−𝑎+1 Quiz 5:
𝑎+𝑏
2
𝑁 (𝑥 −µ)2 Mean: µ = 𝐸(𝑋) = 1. There is no difference between the z distribution and the t distribution.
𝑖 2
σ = ∑ 𝑁 2 False
2 (𝑏−𝑎+1) −1
𝑖=1 Variance: σ = 𝑉(𝑋) = 2
- For the n observations in a random samplede noted by x1, x2, …, xn:
12 2. The 𝑋 distribution and statistic are used to develop confidence intervals
𝑛 (𝑥𝑖−𝑥)
2 around which of the following values? Population variance
2
𝑠 = ∑ - Binomial distribution:
𝑛−1
𝑖=1 𝑥 𝑛−𝑥
PMF: 𝑝(𝑥) = (𝑛𝐶𝑥)𝑝 (1 − 𝑝) (𝑓𝑜𝑟 𝑥 = 1, 2,...) Quiz 7:
Mean: µ = 𝐸(𝑋) = 𝑛𝑝 Two machines are used for filling plastic bottles with a net volume of 16.0
Frequency Distributions 2 ounces. The fill volumes can be assumed to be normal with standard
• A frequency distribution is a compact summary of data, Variance: σ = 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝)
deviation equal to 0.020 and 0.025 ounces. A member of the quality
expressed as a table, graph, or function. engineering staff suspects that both machines fill to the same mean net
• The data is gathered into bins, defined by class intervals. - Poisson distribution:
−λ 𝑥 volume, whether or not this volume is 16.0 ounces. A random sample of 10
• Rule of Thumb: the number of bins should be the square root of the PMF: 𝑝(𝑥) =
𝑒 (λ)
𝑓𝑜𝑟 𝑥 = 1, 2, 3,... bottles is taken from the output of each machine.
𝑥!
sample size, 𝒏.
Mean: µ = 𝐸(𝑋) = λ
2 What type of hypothesis test would be most appropriate for this problem?
Variance: σ = 𝑉(𝑋) = λ
Two Sample Z-Test
x- quantity, λ -rate
Quiz 8:
- Geometric distribution:
𝑥−1
1. A small correlation coefficient, let's say less than 0.05, means there is no
PMF: 𝑝(𝑥) = 𝑝(1 − 𝑝) relationship between the independent and dependent variables. False
Histograms
Mean: µ = 𝐸(𝑋) = 1/𝑝
A histogram is a visual display of a frequency distribution 2. The residual can be found by the following equation: 𝑒𝑖 = 𝑦𝑖 − 𝑦𝑖. False
2 2
Steps to construct a histogram with equal bin widths: Variance: σ = 𝑉(𝑋) = (1 − 𝑝)/𝑝
1) Label the bin boundaries on the horizontal scale. 3. In simple linear regression, the values β1 and β0 are estimates for the
2) Mark and label the vertical scale with the frequencies or relative - Hypergeometric distribution: slope and intercept, respectively, of the line that relates an independent
frequencies. 𝑝(𝑥) =
(𝐷𝐶𝑥)((𝑁−𝐷)𝐶(𝑛−𝑥))
variable to a dependent variable. True
(𝑁𝐶𝑛)
3) Above each bin, draw a rectangle whose height is equal to the frequency 𝑛𝐷 4. Which of the following methods is being used to generate the estimates of
µ=
corresponding to that bin. 𝑁 the coefficients for a linear regression equation? Ordinary Least Squares
2 𝑛𝐷 𝐷 𝑁−𝑛
σ = 𝑁
(1 − 𝑁
)( 𝑁−1 )
Practice Test 1:
Quiz questions: 1. Consider the following function: f(t) = 2kt3, where 2 ≤ t ≤ 4. For what
Quiz 1: value of k does this function become a probability density function?
1. If two events are independent of one another then the probability of their a. 1/136 b. 1/116 c. 1/260 d. 1/120 e. Answer not given
intersection is zero: False
2. An experiment that results in the same outcome when replicated is
Calculator keys (TI-84): referred to as a random experiment: True
Summation: Math -> 0:summation 3. A random variable is said to be discrete if it has an infinite number of
𝑛! values in a finite interval of its range: False
Permutations: Math -> Prob -> nPr (𝑛𝑃𝑟 = )
(𝑛−𝑟)! 4. The combination rule determines the number of ways a selection of x
𝑛!
Combinations: Math -> Prob -> nCr (𝑛𝐶𝑟 = 𝑟!(𝑛−𝑟)!
) items from a set of n available items can happen if we assume order does
Normal Distribution: 2nd -> vars -> normalcdf (use 1EE99 for infinity) not matter: True
