Finding density function $cX$, and what kind of random variable is $cX$.

Question

Given that $X$ is an exponential random variable with parameter $\lambda $, and $c>0$. Finding density function $cX$, and what kind of random variable is $cX$.

Answer 1

Let $F$ be the cdf of $X$. Then $P(cX \leqslant t) = P(X \leqslant \tfrac{t}{c}) = F(\tfrac{t}{c})$ as $c>0$. Thus $F(\tfrac{t}{c})$ is the cdf of $cX$. Can you remember what is the relationship between the cdf and the density? If so, can you find the density now?