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I'm working through Norman Matloff's book 'Probability and statistics for data science', chapter 11 (multivariate distributions).

I'm a bit concerned about an example given.

$X$ and $Y$ are independent and exponentially distributed random variables with mean 2 and mean 1 respectively.

$W$ is a convolution of two exponential densities corresponding to the pdfs of $X$ and $Y$.

The density of W is calculated to be $e^{-0.5t} - e^{-t}$, with $0 < t < \infty$

But this integral does not sum to 1 in the support of $t$. It's divergent. So it can't be a valid pdf.

My question is how should I interpret this example that the authors gave?

He's trying to give an example of convolution. Is this example flawed? If it is flawed, then is there a better example of this principle?

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  • $\begingroup$ Can you elaborate a bit on why you think the integral of this density isn't convergent? $\endgroup$
    – Wei
    Commented Sep 4 at 15:35
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    $\begingroup$ The result is correct and $\int_0^\infty e^{-0.5t}-e^{-t}\,dt=1$. $\endgroup$
    – Sassatelli Giulio
    Commented Sep 4 at 15:43
  • $\begingroup$ Thanks all. Truth be told I plugged it into Wolfram Alpha which told me it doesn't converge. After trying again on paper it finally evaluated to 2-1=1. $\endgroup$
    – mathstatsR101
    Commented Sep 4 at 16:17
  • $\begingroup$ WA agrees that it is $1$...what did you ask it? $\endgroup$
    – lulu
    Commented Sep 4 at 17:27

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