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## Q. 2.9

Use the Dirac delta function to write an expression for the probability density function of the discrete random variable X of Examples 2.1 and 2.4, having

$P (X=1) = P (X=2) = P (X=3) = P (X=4) = P (X=5) = P (X=6) = 1/6$

This probability density function can be written as

## Verified Solution

Strictly speaking, $\delta (\cdot )$ is not a function, but it can be thought of as a limit of a sequence of functions, as considered in Appendix A.

$p_{X} (u) =\frac{1}{6} [\delta (u-1) + \delta (u-2)+\delta (u-3)+\delta (u-4)+\delta (u-5)+\delta (u-6)]$

This expression is exactly the formal derivative of the cumulative distribution function in Example 2.4.