Difference between revisions of "ECE600 F13 Conditional Distributions for Two Random Variables mhossain" - Rhea

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Conditional Distributions

There are many applications of probability theory where we want to know the probabilistic behavior of a random variable Y given the value of another random variable X. We get this using conditional distributions.

Definition $ \qquad $ For random variables X and Y defined on (S,F,P), the joint cdf of X and Y given an event M ∈ F, with P(M) >0, is

A special case of great interest is:

where x$ _1 $ < x$ _2 $. In this case, we have, assuming P(x$ _1 $ < X ≤ x$ _2 $),

What we really want is f_$ _Y $(y|x$ _1 $ < X ≤ x$ _2 $), so we need to differentiate with respect to y. We do not have a name for this partial differentiation and we have not talked about how to find it, but we will need it here.

Now, writing

we have

But what we really want is f$ _Y $(y|X = x) ∀x ∈ R. We cannot set x$ _1 $ = x$ _2 $ in the above equation, so instead, let

Setting x$ _1 $ = x and x$ _2 $ = x + Δx,

Multiplying the denominator and numerator by 1/Δx and taking the limit,

Now let

The numerator is the derivative of F(x) with respect to x and the denominator is the derivative of F$ _X $(x) with respect to x, so

Similarly,

Notation

Writing two equations above in terms of f$ _{XY} $ and setting them equal to each other gives Bayes' formula:

We also have a Total Probability Law:

So we can write f$ {X|Y} $(x|y) in terms of f$ {Y|X} $(x|y)f$ _X $(x), which can be very useful.

Note that if X and Y are independent, we have

So f$ {Y|X} $(x|y) does not depend on x.

Summary of three forms of Bayes' formula that we have derived:

$ \bullet P(A|B)=\frac{P(B|A)P(A)}{P(B)}\qquad A,B\in\mathcal F $

Use this form when X and y are discrete with A = {Y = y}, B={X = x}, so

$ \;p_{Y|X}(y|x) =\frac{p_{X|Y}(x|y)p_Y(y)}{p_X(x)} $

where p$ _{Y|X} $(y|x) ≡ P(Y=y|X=x) and p$ _{X|Y} $(x|y) ≡ P(X=x|Y=y) are conditional pdfs.

$ \bullet P(M|Y=y)=\frac{f_Y(y|M)P(M)}{f_Y(y)}\qquad M\in\mathcal F $

Use this when Y is continuous and X is discrete with M = {X = x}, so

$ \;p_{X|Y}(x|y) =\frac{f_{Y|X}(y|x)p_X(x)}{f_Y(y)} $

$ \bullet f_{Y|X}(y|x)=\frac{f_{X|Y}(x|y)f_Y(y)}{f_X(x)} $

Use this when X,Y are both continuous.

References

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

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