Lecture 16

Conditional Distributions and Independence

Grace Tompkins

Last modified — 13 Nov 2026

Learning Outcomes

By the end of this lecture, students are anticipated to be able to:

Calculate and utilize conditional distributions
Determine whether random variables are independent based on their conditional and joint distributions

1 Conditional Distributions

Conditioning on Discrete RVs

Let \(Y\) be a random variable, where \(X\) is a discrete random variable. Suppose that \(\mathbb{P}(X = x) >0\) for some \(x\). Then the conditional distribution of \(Y\) given \(X = x\) assigns to each set \(A \subset {\mathbb{R}}\) the probability \[\mathbb{P}(Y \in A \ \vert\ X = x) = \frac{\mathbb{P}(Y \in A, X = x)}{\mathbb{P}(X = x)}.\]

Here, we do not specify whether \(Y\) is continuous or discrete.

Conditioning on Discrete RVs

If \(X\) and \(Y\) are both discrete random variables, then the conditional PMF of \(Y\) given \(X = x\) is defined by \[p_{Y|X}(y \ \vert\ x) = \frac{\mathbb{P}(Y = y, X = x)}{\mathbb{P}(X = x)} = \frac{p_{X,Y}(x,y)}{p_X(x)}.\]

Note: we can rearrange this equation to get \[p_{X,Y}(x,y) = p_{Y|X}(y \ \vert\ x)p_X(x)\]

Conditioning on Discrete RVs

Let \(X\) and \(Y\) be the results of rolling two fair 6-sided dice.
Let \(V = X + Y\) and \(W = \max\{X, Y\}\).

The joint PMF of \(W\) and \(V\) is

\(W\ \backslash\ V\)	2	3	4	5	6	7	8	9	10	11	12
1	1/36	0	0	0	0	0	0	0	0	0	0
2	0	2/36	1/36	0	0	0	0	0	0	0	0
3	0	0	2/36	2/36	1/36	0	0	0	0	0	0
4	0	0	0	2/36	2/36	2/36	1/36	0	0	0	0
5	0	0	0	0	2/36	2/36	2/36	2/36	1/36	0	0
6	0	0	0	0	0	2/36	2/36	2/36	2/36	2/36	1/36

To condition on \(W\), we look at a particular row, while to condition on \(V\), we look at a particular column. Then renormalize by the sum of the row/column.

Conditioning on Discrete RV

\(W\ \backslash\ V\)	2	3	4	5	6	7	8	9	10	11	12
1	1/36	0	0	0	0	0	0	0	0	0	0
2	0	2/36	1/36	0	0	0	0	0	0	0	0
3	0	0	2/36	2/36	1/36	0	0	0	0	0	0
4	0	0	0	2/36	2/36	2/36	1/36	0	0	0	0
5	0	0	0	0	2/36	2/36	2/36	2/36	1/36	0	0
6	0	0	0	0	0	2/36	2/36	2/36	2/36	2/36	1/36

Given the above table, what is \(\mathbb{P}(W = 5 \ \vert\ V = 8)\)

Conditioning on Discrete RVs

Conditioning on Continuous RVs

If \(X\) and \(Y\) are jointly absolutely continuous random variables, then the conditional density of \(Y\) given \(X = x\), is the function \[f_{Y|X}(y \ \vert\ x) = \frac{f_{X,Y}(x,y)}{f_X(x)},\] valid for any \(y\in{\mathbb{R}}\), and all \(x\) such that \(f_X(x) > 0\).

Note: we can rearrange this equation to get \[f_{X,Y}(x,y) = f_{Y|X}(y \ \vert\ x)f_X(x)\]

Conditioning on Continuous RVs

Let \(X\) and \(Y\) be jointly absolutely continuous random variables with joint PDF \(f_{X,Y}\). The conditional distribution of \(Y\) given \(X = x\) assigns to each set \(A \subset {\mathbb{R}}\) the probability \[\mathbb{P}(a \le Y \le b \ \vert\ X = x) = \int_a^b f_{Y|X}(y\ \vert\ x)\, \mathsf{d}y,\] valid for all \(x\) such that \(f_X(x) > 0\).

Conditioning on Continuous RVs

Let \(X\) and \(Y\) be jointly continuous random variables with joint PDF \[f_{X,Y}(x,y) = \frac{1}{x} e^{-x} I_{\{0 \le y \le x\}}(x,y).\]

Find the marginal PDF of \(X\). What distribution does \(X\) have?
Find the conditional PDF of \(Y\) given \(X = x\). What distribution does \(Y \ \vert\ X\) have?

Conditioning on Continuous RVs

Let \(X\) and \(Y\) be independent with \(X \sim {\mathrm{Exp}}(1)\) and \(Y\sim {\mathrm{Exp}}(2)\). Let \(S = X + Y\). Find the conditional density \(f_{X|S}(X = x | S = s)\)

Conditioning on Continuous RVs

Conditional Distributions and Independence

Recall that if \(X\) and \(Y\) are independent, we have: \(\mathbb{P}(Y \in A \ \vert\ X = x) = P(Y \in A)\) This implies:

\[p_{Y|X}(y \ \vert\ x) = p_Y(y)\] and \[f_{Y|X}(y \ \vert\ x) = f_Y(y)\] when \(X\) and \(Y\) are independent.

Conditional Distributions and Independence

Heuristics Types of Distributions of Random Variables

Suppose that \(X\) and \(Y\) have a joint distribution \(\mathbb{P}(X \in A, Y \in B)\).
We can find the joint CDF, PMF, or PDF of \(X\) and \(Y\).
The marginal distribution of \(X\) is the distribution of \(X\) when we “ignore” \(Y\).
The conditional distribution of \(X\) given \(Y = y\) is the distribution of \(X\) when we “fix” \(Y\) to be \(y\).
The marginal comes from “summing out” or “integrating out” the other variable along the rows or columns of the table.
The conditional comes from “dividing out” the other variable. Fix a row or a column, and then renormalize (divide out).
When we say \(Y\ \vert\ X=x\), we usually specify a formula. Meaning, we write down a function that gives the conditional PMF or PDF of \(Y\) given \(X = x\) for all \(x\) such that \(p_X(x) > 0\) or \(f_X(x) > 0\). Renormalizing the row of a table gives the conditional PMF for a specific value of \(x\).

Checkpoint 5

You may have:

HANDWRITTEN Cheat Sheet
Non-graphing, non-programmable calculator
Writing materials

Phones and laptops put away, please! You can have 20 minutes.

Reminders:

No class Monday November 9. No checkpoint on Friday, either!
Midterm 2 is on Friday November 20