The question is as follows:

Suppose that X1 and X2 are independent, identically distributed exponential random variables. Determine the PDF for for X1 - X2.

I understand that because X1 and X2 are IID, they have the same parameter and the same distribution. In this case, it is an exponential distribution. I am also under the assumption that I will need to use some variant of the PDF of the regular distribution:

Other than this, I am not certain how to solve this. Thank you.

  • $\begingroup$ do you know about moment-generating functions? $\endgroup$ – symplectomorphic Apr 30 '14 at 0:42
  • $\begingroup$ also, see math.stackexchange.com/questions/115022/… $\endgroup$ – symplectomorphic Apr 30 '14 at 0:43
  • $\begingroup$ I am familiar with moment generating functions but would like to see some more examples. Let me check the link out that you provided. Ty. $\endgroup$ – DaGr8Gatzby Apr 30 '14 at 0:45

You can let u = x1-x2, and v = x1, and then perform the transformation. Since x1 and x2 are iid, f(x1,x2)=f(x1)*f(x2).

The steps are:

  1. find the range of u and v
  2. get the jacobian value
  3. find x1=a(u,v), x2=b(u,v) , a,b are functions (i.e. x1=v and x2=v-u)
  4. then the joint density function g(u,v)=f(a(u,v),b(u,v))
  5. then find the marginal density function g(u) since u=x1-x2 is what you want.
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