STAT 400 Chap.3 Discrete Random Variables and Their Moments
Discrete Random Variables and Their Moments
From outcomes to numbers
A random variable attaches a number to each outcome, which lets probability questions become algebra. A discrete random variable takes a finite or countably infinite set of values, called its support.
Its probability mass function gives the probability of each value, and the course asks for three checks whenever a pmf is proposed: every value on the support is positive, the values sum to one, and the probability of a set is the sum of the masses in it.
When the support is infinite, the second check is a series, exactly as in the first week.
The CDF and interval probabilities
The cumulative distribution function F(x) = P(X ≤ x) is a staircase for a discrete variable: flat between support points and jumping by the mass at each one. Interval probabilities come from differences of F, and for a discrete variable the endpoints matter.
P(a ≤ X ≤ b) and P(a < X ≤ b) differ by the mass at a, so writing the event carefully before subtracting is the habit that prevents lost marks.
Expectation and variance
The expected value is the probability-weighted average of the support, and the rule for the expectation of a function lets you compute E[X²] or E[1/X] without finding a new distribution. The variance measures spread about the mean.
The shortcut Var(X) = E[X²] − μ² is the form to use under time pressure, and it doubles as an error check, since a negative result means one of the two moments is wrong.
Linear transformations follow two short rules: adding a constant shifts the mean and leaves the variance alone, while multiplying by a constant multiplies the mean by it and the variance by its square.
Moment generating functions
The moment generating function M(t) = E[e^(tX)] stores every moment of a distribution. Its first derivative at zero is the mean and its second derivative at zero is E[X²].
It also identifies a distribution uniquely: for a finite support, the coefficient of each exponential term is the probability of the matching value.
The course uses the mgf both ways, from a pmf to moments and from a recognisable mgf back to the named distribution, and later uses it to explain why sums of independent normal variables stay normal.
Why it matters for Exam 1
This chapter closes the first content block in the calendar, so expect questions that combine it with the earlier probability rules, such as finding a pmf from a counting argument and then its mean and variance.
What this chapter covers
- 01
Support and probability mass functions
- 02
Cumulative distribution functions
- 03
Expected value of a function
- 04
Variance and the shortcut formula
- 05
Linear transformations
- 06
Moment generating functions
Worked example · free
Moments from a small pmf
- 2E[X] = 1(0.5) + 2(0.3) + 4(0.2) = 0.5 + 0.6 + 0.8 = 1.9.
- 2E[X²] = 1(0.5) + 4(0.3) + 16(0.2) = 4.9, so Var(X) = 4.9 − 1.9² = 1.29.
- 2E[Y] = 3(1.9) − 1 = 4.7 and Var(Y) = 9(1.29) = 11.61; the constant −1 does not affect spread.
Key terms
- Random Variable
- A function that assigns a real number to each outcome of an experiment, so events can be described by values.
- Probability Mass Function
- The function f(x) = P(X = x) for a discrete variable, positive on the support and summing to one.
- Cumulative Distribution Function
- The function F(x) = P(X ≤ x), which rises from 0 to 1 and, for a discrete variable, jumps at each support point.
- Expected Value
- The probability-weighted average of the values of a random variable, the balance point of its distribution.
- Moment Generating Function
- The function M(t) = E[e^(tX)], whose derivatives at zero give the moments and whose form identifies the distribution.
Discrete Random Variables and Their Moments FAQ
Why is E[X²] not equal to the square of E[X]?
Squaring and averaging do not commute: averaging the squares weights large values more heavily than squaring the average does. The difference E[X²] − (E[X])² is exactly the variance, which is positive whenever X is not constant.
How do I get a pmf back from a moment generating function?
For a finite support, write the mgf as a sum of terms of the form c·e^(tx). Each coefficient c is the probability of the value x in the exponent. For a familiar form, match it to a known mgf instead.
What happens to the variance when I multiply a random variable by a negative number?
The variance is multiplied by the square of the constant, so it stays nonnegative. A negative multiplier flips the direction of the distribution but does not change its spread except through the squared factor.
Exam move
For each new pmf, fill the four-column table and verify the total probability before computing any moment. Practise one mgf question in each direction: derivatives to moments, and a recognised mgf back to a named distribution.
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