What is the distribution of $Y=\frac{P_1X}{N_0}$ given $X\sim E(1/\sigma^2)$?

Question

$X$ is an exponential random variable with parameter $\frac{1}{\sigma^2}$.

If $Y=\frac{P_1X}{N_0}$ why is it exponentially distributed with parameter $\frac{N_0}{P_1\sigma^2}$?

Answer 1

For questions where you transform variables, it is often useful to look at the CDF of the r.v. rather than the PDF.

If $X$ is exponentially distributed with parameter $1/\sigma^2$, then $$P(X\le x)=1-e^{-x/\sigma^2}$$Therefore we get$$P(Y\le y)=P\left(X\le \frac{N_0y}{P_1}\right)=1-\exp\left(-\frac{N_0 y}{P_1\sigma^2}\right)=1-\exp\left(-\frac{N_0}{P_1\sigma^2} y\right)$$This is the CDF for an exponential variable with parameter $\frac{N_0}{P_1\sigma^2}$. So $Y$ is distributed exponentially with this parameter.