Symbol Meaning Notes / Usage
Random variables Variables whose values arise from chance
experiments
i-th observation One data point in a sample
Sample size Number of observations collected
Population size Size of entire population
Population mean True average of population
Sample mean Estimate of population mean
Population variance
Sample variance
Standard deviation Square root of variance (pop / sample)
Population / sample proportion For categorical data tests
Test statistics Used for z-, t-, chi-square, and F-tests
Significance level Probability of Type I error (usually 0.05)
Probability of Type II error 1 − Power
Population / sample correlation Strength of linear association
Expected value (mean) Long-run average of X
Variance of X
Covariance of X and Y
Probability density function For continuous distributions
Probability of event A
Critical values From Z or t tables
Degrees of freedom Often
Poisson mean rate Expected events per interval
,0. Distributions
Discrete Distributions:
Distribution / Mathematical Expression Conditions / Need-to-Know
Concept
Binomial • Number of successes in n independent trials.• Each trial
Distribution ( has two outcomes (success/fail).• Mean , Var
)
Poisson • Counts events in a fixed interval.• Events independent and
Distribution ( ) occur at constant rate .• Mean = Var = .• Approx.
Binomial if .
Discrete Uniform
• All values equally likely.• Mean , Var
( a,…,b )
.
Continous distributions:
Distribution / Mathematical Expression Conditions / Need-to-Know
Concept
Normal • Bell-shaped, symmetric about μ.• Total area = 1.• 68-95-
Distribution ( 99.7 rule for 1-2-3 σ.
)
Standard Normal • Mean 0, SD 1.• Use Z-tables for probabilities.• If
(Z) .
t-Distribution ( ) • Used when σ unknown and n small.• Heavier tails than
Normal.• df . As n→∞ → Z.
Chi-Square • Sum of squared standard normals.• Right-skewed; mean =
Distribution (k) k, var = 2k.• Used in Goodness-of-Fit and Independence tests.
F-Distribution ( • Ratio of two independent χ² variances.• Used in ANOVA and
variance tests.• df₁ = numerator, df₂ = denominator.
Continuous • Constant density over [a,b]; area = 1.• Mean , Var
Uniform ( a,b )
.
0.5 A partial relative frequency distribution is given as shown in the figure. The total sample size
is 200. What is the frequency of class D? A; .22 B; .18 C; .40 D; .?
o =
o D = 0.20 x 200 = 40
, 0.6 Given the following numbers: 120, 230, 110, 115, 160, 130, 150, 105, 195, 155, 105, 360,
120, 120, 140, 100, 115, 180, 235, 255. What is the 90th percentile?
o 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235,
255, 360 → n = 20
o
o The 18th value = 235 and the 19th = 255.
o
0.7 Which distributor is more consistent? A: 11, 10, 9, 10, 11, 11, 10, 11, 10, 10 B: 8, 10, 13, 7,
10, 11, 10, 7, 15, 12
o
o
0.9 Let a sample space be Ω = 1, 2, 3, 4, 5 and three events be A = {1, 3, 5}, B = {2, 3, 5}, and C =
{2, 4}. Which ones are disjoint events?
o Two events are disjoint (mutually exclusive) if they have no elements in common.
o AC = disjoint
0.11 We choose a new car at random and record its color. Here are the probabilities of the most
popular colors for cars made in North America. What is the probability that a randomly chosen
car has any color other than the listed?
o
0.12 Which of the following statements is correct? (uniform distribution)
o x takes distinct countable values (1, 2, 3, 4)
o For a uniform distribution, all probabilities must be equal. Here, f(x) values are di^erent
(1/10, 2/10, 3/10, 4/10)
0.13 Consider a die-rolling experiment with A = {2, 4, 6} and B = {4, 5, 6}. What is the value of (A ∩
B) U (Ac ∩ B)?
o Because — the parts of B that overlap with A and those that
don’t.
0.14 The IT manager of the HvA reports that a computer system experienced 3 component
failures during the past 100 days. What is the probability of no failures in a given day?
o (X= number of events, k= specific number of events for P)
o
Random variables Variables whose values arise from chance
experiments
i-th observation One data point in a sample
Sample size Number of observations collected
Population size Size of entire population
Population mean True average of population
Sample mean Estimate of population mean
Population variance
Sample variance
Standard deviation Square root of variance (pop / sample)
Population / sample proportion For categorical data tests
Test statistics Used for z-, t-, chi-square, and F-tests
Significance level Probability of Type I error (usually 0.05)
Probability of Type II error 1 − Power
Population / sample correlation Strength of linear association
Expected value (mean) Long-run average of X
Variance of X
Covariance of X and Y
Probability density function For continuous distributions
Probability of event A
Critical values From Z or t tables
Degrees of freedom Often
Poisson mean rate Expected events per interval
,0. Distributions
Discrete Distributions:
Distribution / Mathematical Expression Conditions / Need-to-Know
Concept
Binomial • Number of successes in n independent trials.• Each trial
Distribution ( has two outcomes (success/fail).• Mean , Var
)
Poisson • Counts events in a fixed interval.• Events independent and
Distribution ( ) occur at constant rate .• Mean = Var = .• Approx.
Binomial if .
Discrete Uniform
• All values equally likely.• Mean , Var
( a,…,b )
.
Continous distributions:
Distribution / Mathematical Expression Conditions / Need-to-Know
Concept
Normal • Bell-shaped, symmetric about μ.• Total area = 1.• 68-95-
Distribution ( 99.7 rule for 1-2-3 σ.
)
Standard Normal • Mean 0, SD 1.• Use Z-tables for probabilities.• If
(Z) .
t-Distribution ( ) • Used when σ unknown and n small.• Heavier tails than
Normal.• df . As n→∞ → Z.
Chi-Square • Sum of squared standard normals.• Right-skewed; mean =
Distribution (k) k, var = 2k.• Used in Goodness-of-Fit and Independence tests.
F-Distribution ( • Ratio of two independent χ² variances.• Used in ANOVA and
variance tests.• df₁ = numerator, df₂ = denominator.
Continuous • Constant density over [a,b]; area = 1.• Mean , Var
Uniform ( a,b )
.
0.5 A partial relative frequency distribution is given as shown in the figure. The total sample size
is 200. What is the frequency of class D? A; .22 B; .18 C; .40 D; .?
o =
o D = 0.20 x 200 = 40
, 0.6 Given the following numbers: 120, 230, 110, 115, 160, 130, 150, 105, 195, 155, 105, 360,
120, 120, 140, 100, 115, 180, 235, 255. What is the 90th percentile?
o 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235,
255, 360 → n = 20
o
o The 18th value = 235 and the 19th = 255.
o
0.7 Which distributor is more consistent? A: 11, 10, 9, 10, 11, 11, 10, 11, 10, 10 B: 8, 10, 13, 7,
10, 11, 10, 7, 15, 12
o
o
0.9 Let a sample space be Ω = 1, 2, 3, 4, 5 and three events be A = {1, 3, 5}, B = {2, 3, 5}, and C =
{2, 4}. Which ones are disjoint events?
o Two events are disjoint (mutually exclusive) if they have no elements in common.
o AC = disjoint
0.11 We choose a new car at random and record its color. Here are the probabilities of the most
popular colors for cars made in North America. What is the probability that a randomly chosen
car has any color other than the listed?
o
0.12 Which of the following statements is correct? (uniform distribution)
o x takes distinct countable values (1, 2, 3, 4)
o For a uniform distribution, all probabilities must be equal. Here, f(x) values are di^erent
(1/10, 2/10, 3/10, 4/10)
0.13 Consider a die-rolling experiment with A = {2, 4, 6} and B = {4, 5, 6}. What is the value of (A ∩
B) U (Ac ∩ B)?
o Because — the parts of B that overlap with A and those that
don’t.
0.14 The IT manager of the HvA reports that a computer system experienced 3 component
failures during the past 100 days. What is the probability of no failures in a given day?
o (X= number of events, k= specific number of events for P)
o
