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Probability · Lecture 19 of 76 · 15:26

PMF of a Function of a Random Variable

PMF of a Function of a Random Variable on YouTube

Study guide

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?

Vocabulary

normalizing constant (noun)
A number chosen so that a set of values adds up to 1.
The unknown constant a is found so the PMF is properly normalized.
symmetry (noun)
A pattern where parts mirror each other around a central point.
The PMF's symmetry means matching values have equal probability.
solve for (phrasal verb)
To find the value of an unknown by working through an equation.
You solve for a using the fact that the PMF sums to 1.
map to (phrase)
To be linked to or turned into a corresponding value.
Both X equals 2 and X equals negative 2 map to Z equals 4.
unknown constant (noun)
A fixed value in a formula that must be found before the formula can be used.
The unknown constant a must be found before computing probabilities.
elsewhere (adverb)
In any other place not specifically mentioned.
The PMF is zero elsewhere, outside the given values.
function of a random variable (noun)
A new random variable created by applying a rule to an existing one.
Z is a function of the random variable X, since Z equals X squared.
probability mass (noun)
The amount of probability assigned to a specific value.
The probability mass sitting over x equals 2 must be included.
plot (symmetry) (verb)
To draw a graph in order to see a pattern like symmetry.
Plotting the PMF shows its symmetry around zero.
chapter progression (noun)
The planned order in which topics build on each other through a course.
This follows the chapter progression toward algebraic PMF formulas.
algebraically (adverb)
Using symbols and equations rather than pictures or examples.
The PMF of Z can eventually be represented algebraically.
possible values (noun)
The full list of values a random variable is allowed to take.
X's possible values are -3, -2, -1, 1, 2, and 3.
task (noun)
A specific piece of work to be completed.
The second task is deriving the PMF of Z.
only value (phrase)
The single unique answer that satisfies all the conditions.
28 is the only value of a that produces a valid PMF.
outcome (of X) (noun)
One specific value that a random variable can take.
Each outcome of X has its own probability mass.
recitation problem (noun)
A practice exercise used in a smaller class session to apply lecture ideas.
This recitation problem practices PMF manipulation.
second part (noun)
The later section of a multi-part problem.
The second part asks for the PMF of Z equals X squared.
represent (algebraically) (verb)
To describe a relationship using symbols and formulas.
The PMF of Z is eventually represented algebraically.
given values (noun)
The specific numbers or facts stated as part of a problem.
X takes the given values -3, -2, -1, 1, 2, and 3.
gives (a random variable) (verb)
To define or specify a random variable's rule or values.
The problem gives a random variable X with a stated PMF.

Chapters

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Katie Szeto

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
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