QMB3200-Random Variables and Distributions, Sampling and the Central Limit Theorem, and Hypothesis Testing

Random variable

A numeric description of the outcome of an experiment.

Discrete vs. continuous

Discrete random variable: takes specific values, so you find the probability at a value. Continuous random variable: takes values within an interval, so probability is the area under the curve over an interval, and P(X = x) = 0 for any single value.

Expected value (mean)

E(x) = μ = Σ x·f(x). For a binomial, E(x) = np.

Variance

Var(x) = σ² = Σ (x − μ)²·f(x).

Bivariate distribution

Involves two random variables. Covariance and the correlation coefficient measure their linear association. Cov(x,y) = [Var(x+y) − Var(x) − Var(y)] / 2.

Distributions

Binomial, Hypergeometric, Poisson, Uniform, Normal, Standard normal, and exponential

Binomial

f(x) = C(n,x)·pˣ·(1 − p)^n-x. E(x) = np. Var(x) = np(1 − p) = npq, where q = 1 − p. Excel: =BINOM.DIST(x,n,p,TRUE) is cumulative; =BINOM.DIST(x,n,p,FALSE) is the probability of exactly x.

Hypergeometric

Similar to the binomial, but the trials are not independent (sampling without replacement), so the probability of success changes from trial to trial.

Poisson

Estimates the number of occurrences over a specified interval of time or space. f(x) = μˣ·e^-μ / x!, where μ is the mean number of occurrences in the interval.

Uniform

All possible values are equally likely, such as rolling a die once.

Normal

The bell curve. The most widely used distribution for continuous random variables. It also describes the likely results obtained through sampling.

Standard normal

A bell curve with the mean set to 0 and the standard deviation set to 1 (use z-scores). Excel: =NORM.S.DIST(z,TRUE).

Exponential

A continuous distribution. f(x) = (1/μ)·e^(−x/μ), where μ is the mean (rate λ = 1/μ).

Random sample

A sample chosen by a random process.

Matched (paired) samples

Observations that are meaningfully paired, such as measurements from the same subjects before and after a treatment. Analyze the differences within each pair.

Central limit theorem

Under appropriate conditions, the sampling distribution of the sample mean becomes approximately normal as the sample size increases.

Standard error

The standard deviation of the sample mean: σ / √n (when σ exists and the observations are independent).

Larger sample size

Usually reduces the standard error and the sampling variability. It does not guarantee that every estimate is closer to the true value, and it does not remove bias.

Null hypothesis H₀

A statement about a population parameter that is tested. Usually contains equality. Example: H₀: μ = 24.

Alternative hypothesis H^a

The competing statement, set by the research question: μ ≠ 24 (two-tailed), μ > 24 (upper-tailed), or μ < 24 (lower-tailed).

Test statistic

A number calculated from the sample data and compared with a reference distribution or critical value to judge the evidence against H₀.

p-value and α

If p ≤ α, reject H₀. If p > α, fail to reject H₀. Failing to reject does not prove H₀ true. (Some courses use p < α; follow your instructor.)

Type I error

Rejecting a null hypothesis that is actually true.

Type II error

Failing to reject a null hypothesis that is actually false.

Tests about a mean

T-test, Matched-pair t-test, z procedure, t procedure

T-test

A family of tests about population means: one-sample, independent two-sample, and paired-sample. The correct one depends on the design and the question.

Matched-pair t-test

Tests whether the population mean of the paired differences is zero (or another specified value).

z procedure

Used when the population standard deviation σ is known: z = (x̄ − μ₀) / (σ / √n).

t procedure

Used when σ is unknown and estimated by the sample standard deviation s: t = (x̄ − μ₀) / (s / √n), with n − 1 degrees of freedom for a one-sample t-test. These are

introductory rules; assumptions and study design still matter.