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## Q. 2.6

Find the probability density function for the random variable X of Example 2.2, for which

 $F_{Y}(u)=0$ For $u \lt 0$ $F_{Y}(u)=0.05(u+4)$ For $0 \leq u \lt 10$ $F_{X}(u)=1$ For $u \geq 10$

## Verified Solution

Differentiating the cumulative distribution function, the probability density function is found to be

 $p_{X}(u)=0.1$ For $0 \leq u \lt 10$ $p_{X}(u)=0$ otherwise

or, equivalently,

$p_{X}(u)=0.1U(u)U(10-u)$           or             $p_{X}(u)=0.1[U(u)-U(u-10)]$

Note that the final two forms differ only in their values at the single point $u=10$. Such a finite difference in a probability density function at a single point (or a finite number of points) can be considered trivial, because it has no effect on the cumulative distribution function, or any other integral of the density function.