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The Comer Lectures on Random Variables and Signals

Slectures by Maliha Hossain


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. 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

$ 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, define

$ 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 ≤ a} for some a ∈ R. Then

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

Two cases:

  • Case (i): $ x > a $
$ F_X(x|B) = \frac{P(X\leq a)}{P(X\leq a)} = 1 $
  • Case (ii): $ x < a $
$ F_X(x|B) = \frac{P(X\leq x)}{P(X\leq a)} = \frac{F_X(x)}{F_X(a)} $


Fig 1: {X ≤ x} ∩ {X ≤ a} for the two different cases.


Now,

$ f_X(x|B) = f_X(x|X\leq a)=\begin{cases} 0 & x>a \\ \frac{f_X(x)}{F_X(a)} & x\leq a \end{cases} $
Fig 2: f$ _X $(x) and f$ _X $(x$ | $X ≤ a).


Bayes' Theorem for continuous X:
We can easily see that

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

from previous version of Bayes' Theorem, and that

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

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 AF. We could define this as

$ P(A|X=x)\equiv\frac{P(A\cap \{X=x\})}{P(X=x)} $

but the right hand side (rhs) would be 0/0 since X is continuous.
Instead, we will use the following definition in this case:

$ P(A|X=a)\equiv\lim_{\Delta x\rightarrow 0}P(A|x<X\leq x+\Delta x) $
$ \Delta x > 0 \ $

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

$ \begin{align} P(A|X=x) &= \lim_{\Delta x\rightarrow 0}\frac{P(x<X\leq x+\Delta x|A)P(A)}{P(x<X\leq x+\Delta x)} \\ \\ &= P(A)\lim_{\Delta x\rightarrow 0}\frac{F_X(x+\Delta x|A)-F_X(x|A)}{F_X(x+\Delta x)-F_X(x)} \\ \\ &= P(A)\frac{\lim_{\Delta x\rightarrow 0}\frac{F_X(x+\Delta x|A)-F_X(x|A)}{\Delta x}}{\lim_{\Delta x\rightarrow 0}\frac{F_X(x+\Delta x)-F_X(x)}{\Delta x}}\\ \\ &=P(A)\frac{f_X(x|A)}{f_X(x)} \end{align} $

So,

$ P(A|X=x)=\frac{f_X(x|A)P(A)}{f_X(x)} $

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

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



References



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