Lecture 5: Discrete Random Variables I

What this lecture covers

This lecture opens a new unit on random variables, which assign numerical values to the outcomes of a probabilistic experiment. It defines random variables formally as functions from the sample space to the real numbers, introduces the probability mass function (PMF) as a way to describe how likely each numerical value is, and then develops the central concept of expected value, along with the expected value rule for functions of random variables and the concept of variance.

After watching, you should be able to define a random variable and its PMF, compute PMFs for examples like the geometric and binomial random variables, calculate expected values directly or by exploiting symmetry, apply the expected value rule to compute expectations of functions of a random variable without deriving their PMF, and compute and interpret variance, including how it transforms under linear changes of variable.

Key ideas

• Random variable: formally a function from the sample space to the real numbers, not a "random" or "variable" quantity in the everyday sense; capital letters denote the random variable, lowercase letters denote its numerical values.
• Functions of random variables are random variables: if Y = g(X) for a random variable X, then Y is itself a random variable, since the outcome of the experiment determines X and hence Y.
• Probability mass function (PMF): p_X(x) is the probability that X takes the value x; PMF values are non-negative and sum to 1 over all possible x.
• Geometric PMF: for the number of coin tosses until the first head, p_X(k) = (1-p)^(k-1) * p, decreasing geometrically as k grows.
• Expected value: E[X] = sum of x * p_X(x), interpretable as a long-run average under frequency interpretation of probability, or as the center of gravity of the PMF's bar graph.
• Expected value rule (law of the unconscious statistician): E[g(X)] = sum of g(x) * p_X(x), letting you compute the expectation of a function of X without first finding the PMF of g(X).
• Linearity of expectation: E[alpha*X + beta] = alpha*E[X] + beta, an important exception to the general rule that averaging and applying a function do not commute.
• Variance: Var(X) = E[(X - E[X])^2], a measure of how spread out a distribution is; it is always non-negative, unaffected by adding a constant, and scales by alpha^2 when X is multiplied by a constant alpha.

Walkthrough

Defining random variables (2:04)

Using the example of measuring a random student's height, the lecture defines a random variable as a function from outcomes in the sample space to real numbers. It emphasizes that a single experiment can involve multiple random variables (height, weight, height in centimeters), and that a function of a random variable is itself a random variable, since the outcome determines everything downstream.

Discrete vs. continuous random variables and the PMF (7:11)

The lecture distinguishes discrete random variables (taking a countable set of values, such as rounded heights) from continuous ones, and explains why the course starts with the discrete case. It then defines the PMF p_X(x) as the probability the random variable equals x, states the two defining properties (non-negativity and summing to 1), and works through the geometric PMF for the number of tosses until the first head.

Computing PMFs by counting outcomes (18:43)

Two more PMF examples are worked: the minimum of two tetrahedral die tosses, and the number of heads in n coin tosses (the binomial case from the previous lecture, now expressed with PMF notation p_X(k) = (n choose k) p^k (1-p)^(n-k)). The lecture notes that for large n the binomial PMF's bar graph approaches a bell-shaped curve, a preview of later material.

Expected value (24:54)

The expected value is introduced as a probability-weighted average, illustrated with a payoff game, and given a second interpretation as the center of gravity of the PMF's bar graph. This center-of-gravity intuition is used to quickly find the expected value of a symmetric uniform random variable without algebra.

The expected value rule for functions of a random variable (30:59)

To compute E[g(X)], the lecture shows two equivalent ways of summing: over outcomes grouped by the value of Y = g(X), or directly over the values of X, weighting each g(x) by p_X(x). The second approach avoids computing the PMF of Y, and is called the law of the unconscious statistician. The lecture stresses that, in general, E[g(X)] does not equal g(E[X]).

Linearity of expectation (36:16)

As a key exception to that general rule, the lecture proves E[alpha*X + beta] = alpha*E[X] + beta using the expected value rule, illustrating it with the conversion of height from inches to centimeters plus a constant bonus.

Variance (42:24)

Variance is defined as the expected squared deviation from the mean, E[(X-E[X])^2], measuring how spread out the PMF is. The lecture notes the equivalent computational formula using E[X^2] and E[X], and derives how variance behaves under linear transformations: unaffected by adding a constant, scaled by alpha^2 when multiplying by alpha.

Before you watch

• Review the counting techniques and binomial probability formula from the previous lecture, since the binomial PMF example builds directly on them.
• Be comfortable with basic algebraic manipulation of sums, since PMF and expectation calculations involve summing over possible values.

Check your understanding

1. Why is a random variable formally defined as a function rather than as a number?
2. How does the expected value rule let you compute E[g(X)] without finding the PMF of g(X) first?
3. Why does E[g(X)] generally not equal g(E[X]), and what is the important exception?
4. Why is variance unaffected by adding a constant to a random variable, but scaled by alpha^2 when multiplying by a constant alpha?
5. What does the center-of-gravity interpretation of expected value tell you about the expected value of a symmetric PMF?