ECE600 F13 rv conditional distribution mhossain - Rhea

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

P(A|B) = \frac{p(A\cap B)}{P(B)}

∀ 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

P_X(x|B)\equiv P(X=x|B)=\frac{p(\{X=x\}\cap B)}{P(B)}

∀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:

P(A|B)=\frac{P(B|A)P(A)}{P(B)};\quad P(B), P(A)>0

and

P(B)=\sum_{i = 1}^nP(B|A_i)P(A_i)

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

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

p_X(x|B) = \frac{P(B|X=x)p_X(x)}{P(B)}

where $$ p_X(x|B) $$ is the conditional pmf of X given B and $$ p_X(x) $$ is the pmf of X.

We also can use the TPL to get

p_X(x) = \sum_{i=1}^n p_X(x|A_i)P(A_i)

Continuous X

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

F_X(x|B)\equiv P(X\leq x|B) = \frac{P(\{X\leq x\}\cap B)}{P(B)}

as the conditional cdf of X given B.
The conditional pdf of X given B is then

f_X(x|B) = \frac{d}{dx}F_X(x|B)

Note that B may be an event involving X.

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

F_X(x|B) = \frac{P(\{X\leq x\}\cap\{X\leq a\})}{P(X\leq a)}

Two cases:

References

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

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ECE600 F13 rv conditional distribution mhossain - Rhea

Discrete X

Continuous X

References

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