# Convolution of probability mass functions (3 non-parametric distributions)

I am familiar with the convolution of probability mass function when it involves two random variables, but I get a little confused when there's a third one.

I have to find the probability mass function of Z = W + X + Y where W, X and Y are independent random variables and with the given table:

I am not sure about the general formula to solve this. Thank you in advance for your help!

• Hi @boiba, did this answer your question? If so, could you please accept and upvote the answer? Otherwise, what can be clarified? – Arya McCarthy Feb 25 at 14:48

## 1 Answer

You asked for a general formula for convolving several independent variables, so I'll give you that rather than solving the specific case from your homework.

If you can convolve W and X into a third variable (call it Q), then you can convolve Q with Y to get Z, following the same procedure you just used:

$$f_Z(z) = f_Q(z) * f_Y(z) = f_W(z) * f_X(z) * f_Y(z)$$

More details can be found from this tutorial. Note the claim:

Nevertheless, this quickly becomes computationally difficult. Thus, we often resort to other methods if we can. One method that is often useful is using moment generating functions, as we discuss in the next section.

• This answer is incorrect because it implicitly assumes all three variables are independent, whereas the variables in the question are not independent. – whuber Feb 24 at 15:33
• Thank you for pointing that out: I misread the table. It appears to describe three probability distributions on the set $\{0,1,\ldots, 4\}.$ I agree that your answer is appropriate and correct. I would still suggest stating the independence assumption explicitly. (+1). – whuber Feb 24 at 16:12
• Thanks for the clarity! Just updated it. – Arya McCarthy Feb 24 at 16:13