PMF of a Function of a Random Variable

What this lecture covers

This recitation problem gives a random variable X taking values -3, -2, -1, 1, 2, 3 with PMF p_X(x) = x^2/a for an unknown constant a, and asks you to find a, then to find the PMF of Z = X^2, a function of X. It is a practice problem for manipulating PMFs and applying the normalization axiom that a PMF must sum to 1.

After watching, you should be able to use the "PMF values sum to 1" property to solve for an unknown normalizing constant in a PMF, and understand how to approach finding the PMF of a function of a random variable by combining the probability mass of X-values that map to the same output.

Key ideas

• PMF definition: p_X(k) is the probability that random variable X equals k; it can be thought of as a function that returns the probability mass sitting over a given value.
• Normalization axiom: summing a PMF over all possible values of X must equal 1, which gives an equation you can solve for an unknown parameter.
• Symmetry: because p_X(x) = x^2/a depends only on x^2, the PMF is symmetric, so p_X(-3) = p_X(3), p_X(-2) = p_X(2), and p_X(-1) = p_X(1), which simplifies the summation.
• Solving for the unknown constant: summing 2(9/a) + 2(4/a) + 2(1/a) = 28/a and setting it equal to 1 gives a = 28.
• PMF of a function of a random variable: since Z = X^2, several values of X can map to the same value of Z (for example, both X=2 and X=-2 give Z=4), so finding p_Z requires combining the probability mass of all X-values that produce each Z-value.

Walkthrough

Setting up the problem and reviewing what a PMF means (0:01)

The problem gives p_X(x) = x^2/a for x in {-3,-2,-1,1,2,3} and 0 elsewhere, with a > 0 unknown. The lecture reviews that p_X(k) denotes the probability mass lying over the value k, setting up the two tasks: solve for a, then find the PMF of Z = X^2.

Solving for the unknown constant a (2:08)

Using the axiom that a PMF must sum to 1 over all possible values, the lecture plots the PMF (noting its symmetry around 0), sums the six probability masses as 2(9/a) + 2(4/a) + 2(1/a) = 28/a, sets this equal to 1, and solves to get a = 28, showing this is the only value that produces a valid PMF.

Finding the PMF of Z = X^2 (5:10)

The lecture moves on to the second part of the problem, deriving the PMF of Z, which requires identifying which values of X map to each possible value of Z and combining their probabilities, following the chapter progression toward representing p_Z algebraically.

Before you watch

• Review the definition of a probability mass function and the axiom that probabilities of all outcomes in a sample space sum to 1, from earlier lectures in this course.
• Be comfortable with the expected value rule / functions of random variables, since this problem's second part depends on relating X and Z = X^2.

Check your understanding

1. Why must a PMF sum to 1 over all possible values, and how does that let you solve for an unknown parameter?
2. Why does p_X(-3) equal p_X(3) in this problem?
3. When computing the PMF of Z = X^2, why might more than one value of X contribute to the same value of Z?