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Introductory Statistics Unit 4: Random variables & distributions

A random variable assigns a number to each outcome of a chance process. Its distribution tells you the possible values and how likely each one is.

Covers: Discrete & continuous random variables, expected value, binomial distribution

Study guide

A random variable assigns a number to each outcome of a random process. Discrete random variables take countable values (number of heads) and are described by a probability distribution listing each value and its probability, with probabilities adding to 1. Continuous random variables take any value in an interval (time, height), and probabilities are areas under a density curve, so P(X = exact value) = 0.

The expected value (mean) of a discrete random variable is E(X) = Σ x·P(x), the long-run average. Its variance is Σ (x − μ)²·P(x), and the standard deviation is the square root. For linear transformations Y = a + bX, the mean becomes a + bμ and the standard deviation becomes |b|σ. For sums or differences of independent random variables, means add or subtract, but variances always add: Var(X ± Y) = Var(X) + Var(Y).

A binomial setting has four conditions (BINS): Binary outcomes (success or failure), Independent trials, a fixed Number of trials n, and the Same probability of success p each trial. Then P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ, with mean np and standard deviation √(np(1 − p)). Use technology (binompdf/binomcdf) for cumulative probabilities like P(X ≤ 3).

A geometric random variable counts trials until the first success, with P(X = k) = (1 − p)ᵏ⁻¹p and mean 1/p.

A geometric random variable has standard deviation √(1 − p)/p, and P(X > k) = (1 − p)ᵏ, because the first k trials must all be failures. If an observed outcome would be very unlikely, with a probability below about 5%, when an assumed probability is true, that is convincing evidence against the assumption. A likely outcome gives no such evidence.

If X and Y are independent and normal, X + Y and X − Y are also normal. After finding their mean and standard deviation, compute probabilities with z-scores. The standard deviation of a random variable describes how far outcomes typically are from the mean, over many repetitions.

Key ideas

• Discrete random variables have countable values; continuous ones take any value in an interval.
• Expected value is the long-run average.
• Binomial setting (BINS): Binary outcomes, Independent trials, fixed Number of trials, Same probability of success.
• Adding a constant shifts the mean only; multiplying by a constant scales both mean and SD.
• For independent random variables, variances add (never SDs).

Formulas and rules

Common mistakes

• Adding standard deviations instead of variances.
• Using the binomial formula when the trials aren't independent or the count isn't fixed.

Words to know

Expected value of a discrete random variable

E(X) = Σ x·P(x)

Binomial conditions (BINS)

Binary outcomes, Independent trials, fixed Number of trials, Same success probability.

Binomial probability formula

P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ

Mean and SD of a binomial

Mean = np, SD = √(np(1 − p))

Variance of a difference

Var(X − Y) = Var(X) + Var(Y) for independent X, Y.

Linear transformation of a random variable

For Y = a + bX: mean becomes a + b·mean, SD becomes |b|·SD.

Random variable

A numerical outcome of a random process.

Discrete random variable

Takes countable values, each with a probability.

Continuous random variable

Takes any value in an interval; probabilities are areas under a density curve.

Standard deviation of a random variable

√Σ (x − μ)²P(x)

Mean of a sum or difference

μ(X ± Y) = μX ± μY

Geometric random variable

Counts trials until the first success; mean 1/p.

Binomial random variable

Counts successes in n independent trials with the same p.

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.