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A random variable $X$ is said to have an exponential distribution,

$f(x)\ =\ \begin{cases}\lambda e^{-\lambda x}&if\ x>0\\ 0&otherwise\end{cases} $

How can I show that the class of exponentially distributed random variables is closed under multiplication with a positive number

Or in other words, if $X$ is exponentially distributed and $α ∈ (0, ∞)$ how can I show that $ αX$ is also exponentially distributed.

Econfinanceguy
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  • Write down the CDF of $\alpha X$ and differentiate it to see what the pdf is. – geetha290krm Nov 08 '22 at 08:44
  • Does this answer your question? https://math.stackexchange.com/questions/85554/is-the-family-of-exponential-distributions-closed-under-scaling – Alborz Nov 08 '22 at 08:52

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