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.
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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 $\frac{P}{N}E[X]$
Henry
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OK How we drive the CDF and PDF? – BouMokhtar Jul 20 '18 at 13:20
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You know that the mean is $\beta$ and you also know that the mean ist $\frac 1 \lambda$. Now you have $\lambda$ which gives you both the PDF and CDF. – Stefan Jul 20 '18 at 13:30
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So if mean is $\frac{P}{N}E[X]$, the pdf is $\frac{N}{PE[X]}e^{-\frac{Nx}{PE[X]}}$ – BouMokhtar Jul 20 '18 at 13:32