Statistics and Random Variables (Chapters 6–8) - Vocabulary Flashcards

A set of vocabulary flashcards covering key terms and definitions from discrete/continuous random variables, distributions, and the normal/binomial families as presented in chapters 6–8.

Random variable

A numerical description of the outcomes of a random experiment, denoted by X; assigns a number to each result of the experiment.

Discrete random variable

A random variable that can take only a finite or countable set of values (e.g., number of defective items).

Continuous random variable

A random variable that can take any value in an interval; probabilities are assigned to ranges, not exact values (P(X = a) = 0).

Probability distribution (discrete)

A function or table giving P(X = x) for each possible value x of a discrete random variable.

Probability distribution (continuous)

A description of probabilities via a probability density function (PDF) f(x) with P(a ≤ X ≤ b)=∫_a^b f(x)dx.

Probability density function (PDF)

A non-negative function f(x) for a continuous random variable such that ∫ f(x) dx = 1; P(a ≤ X ≤ b)=∫_a^b f(x)dx.

Cumulative distribution function (CDF)

F(x) = P(X ≤ x); a nondecreasing function that aggregates probabilities up to x.

Expected value (mean)

Long-run average value of X; for discrete X, E(X)=∑ xi P(X=xi); for continuous X, E(X)=∫ x f(x) dx.

Variance

A measure of spread: Var(X)=E[(X−μ)²]; for discrete, Var(X)=∑(xi−μ)² P(X=xi); μ is the mean.

Standard deviation

σ, the square root of the variance; σ = √Var(X).

Bernoulli distribution

X takes 1 with probability p and 0 with probability 1−p; mean μ=p and variance σ²=p(1−p).

Binomial distribution

X = number of successes in n independent Bernoulli trials with probability p; X~Binomial(n,p); mean μ=np; variance σ²=np(1−p); pmf P(X=k)=C(n,k)p^k(1−p)^{n−k}.

Uniform discrete random variable

A discrete variable where all possible values have equal probability.

Uniform continuous distribution

A continuous variable X that is equally likely over an interval [a,b], with f(x)=1/(b−a) for a<x<b.

Normal distribution

Bell-shaped, symmetric about μ; X~N(μ,σ²); PDF f(x)=(1/(σ√(2π))) e^{-(x−μ)²/(2σ²)}.

Standard normal distribution

Z ~ N(0,1); obtained by transforming X via Z=(X−μ)/σ to compare across populations.

Z-score

Number of standard deviations a value x is from the mean: z=(x−μ)/σ.

Quantile

Value k such that P(X ≤ k) = p; the p-th percentile of the distribution.

Mode

The most frequently occurring value(s) in a distribution (highest probability).

Median

The value(s) that divide the probability distribution so that half the data lie at or below it.

Mean of Binomial distribution

μ=np, the expected number of successes in n trials with success probability p.

Variance of Binomial distribution

σ²=np(1−p); measures spread of the number of successes in n trials.

Linearity of expectation

E(aX+b)=aE(X)+b for constants a,b; also extends to sums: E[∑ Xi]=∑E[Xi].

Variance under linear transformation

Var(aX+b)=a²Var(X); shifting by b does not affect variance.

Binomial probability mass function

P(X=k)=C(n,k) p^k (1−p)^{n−k} for k=0,1,…,n; used when counting successes in binomial trials.

Probability notation P(X=x)

The probability that the random variable X takes the specific value x in a distribution.

Mean and standard deviation in a normal context

In a normal distribution, about 68% lie within μ±σ, about 95% within μ±2σ, and about 99.7% within μ±3σ.

Inverse normal function (quantiles)

Calculator function used to find values k such that P(X ≤ k) equals a given probability for a normal distribution.

Relation between discrete and continuous descriptions

Discrete: probabilities on distinct values; Continuous: probabilities over intervals via PDFs and integrals.