How to find pdf of exponential random variable given the mean

Question

Let $X$ be a exponential random variable with mean $\beta$.

How we can write the PDF of $X$ given mean $\beta$.

Let $Y$ random variable define by $$ Y=\frac{P}{N}X$$ were $N$ and $P$ are positive real value. What is the PDF and CDF of Y.

Answer 1

Hints:

• An exponential distribution with rate $\lambda$ has pdf $\lambda e^{-\lambda x}$ and CDF $1 - e^{-\lambda x}$ for $x \ge 0$, and has mean $\frac{1}{\lambda}$

• $Y$ is also exponentially distributed, with mean $\frac{P}{N}E[X]$