# Is this not a valid probability density function? [closed]

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?

• Can you elaborate a bit on why you think the integral of this density isn't convergent?
– Wei
Commented Sep 4 at 15:35
• The result is correct and $\int_0^\infty e^{-0.5t}-e^{-t}\,dt=1$. Commented Sep 4 at 15:43
• 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. Commented Sep 4 at 16:17
• WA agrees that it is $1$...what did you ask it?
– lulu
Commented Sep 4 at 17:27