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
,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.
We also can use the TPL to get
Continuous X
Let A = {X≤x}. Then if P(B)>0, B ∈ F, definr
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≤x} for some a ∈ R. Then
Two cases:
References
- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.
