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**The Comer Lectures on Random Variables and Signals**

Topic 7: Random Variables: Conditional Distributions

We will now learn how to represent conditional probabilities using the cdf/pdf/pmf. This will provide us some of the most powerful tools for working with random variables: the conditional pdf and conditional pmf.

Recall that

∀ A,B ∈ *F* with P(B) > 0.

We will consider this conditional probability when A = {X≤x} for a continuous random variable or A = {X=x} for a discrete random variable.

## Discrete X

If P(B)>0, then let

∀x ∈ *R*, for a given B ∈ *F*.

The function p$ _X $ is the conditional pmf of X. Recall Bayes' theorem and the Total Probability Law:

and

if $ A_1,...,A_n $ form a partition of *S* and $ P(A_i)>0 $ ∀i.

In the case A = {X=x}, we get

where p$ _X $(x|B) is the conditional pmf of X given B and $ p_X(x) $ is the pmf of X. Note that Bayes' Theorem in this context requires not only that P(B) >0 but also that P(X = x) > 0.

We also can use the TPL to get

## Continuous X

Let A = {X≤x}. Then if P(B)>0, B ∈ *F*, define

as the conditional cdf of X given B.

The conditional pdf of X given B is then

Note that B may be an event involving X.

**Example:** let B = {X ≤ a} for some a ∈ **R**. Then

Two cases:

- Case (i): $ x > a $

- Case (ii): $ x < a $

Now,

Bayes' Theorem for continuous X:

We can easily see that

from previous version of Bayes' Theorem, and that

if $ A_1,...,A_n $ form a partition of *S* and P($ A_i $) > 0 ∀$ i $, from TPL.

but what we often want to know is a probability of the type P(*A*|*X*=*x*) for some *A*∈*F*. We could define this as

but the right hand side (rhs) would be 0/0 since *X* is continuous.

Instead, we will use the following definition in this case:

using our standard definition of conditional probability for the rhs. This leads to the following derivation:

So,

This is how Bayes' Theorem is normally stated for a continuous random variable X and an event *A*∈*F* with P(*A*) > 0.

We will revisit Bayes' Theorem one more time when we discuss conditional distributions for two random variables.

## References

- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.

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