Practice Problem: What is the conditional density function

Let X be a continuous random variable with probability density function

$ f_X(x)=\left\{ \begin{array}{ll} c x^2, & 1<x<5,\\ 0, & \text{ else}. \end{array} \right. $

Let A be the event $ \{ X>3 \} $. Find the conditional probability density function $ f_{X|A}(x|A). $

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Answer 1

Hint:

Find c by,

$ \int_{-\infty}^{\infty} f_{X}(x)dx =1. $

$ f_{X|A}(x|A)= \frac{f_{X}(x)}{P({X>3})} = \frac{f_{X}(x)}{1- F_{X}(3)} . $ -TA

Comment on Hint

It's important to note that the $ \color{blue} f_X(x) $ given in the final line of the hint is distinct from the pdf given in the problem statement. Specifically, the new $ \color{blue} f_X(x) $ is nonzero only on the range dictated by the occurrence of event 'A' such that

$ \color{blue} f_X(x)=\left\{ \begin{array}{ll} c x^2, & 3<x<5,\\ 0, & \text{ else}. \end{array} \right. $

Note that this is 'new' $ \color{blue} f_X(x) $ is not a valid pdf by itself (violates normalization to 1 axiom), and thus the normalizing denominator is used. -ag

Answer 2

Write it here.

Answer 3

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