Pdf and Coefficient of Kurtosis

1. Dec 19, 2012

Aria1

1. The problem statement, all variables and given/known data

Let X and Y be two independent exponential random variables with a common rate parameter λ>0. Let W=X-Y.
a)Find the pdf of W=X-Y
b) Find the coefficient of kurtosis of W=X-Y

2. Relevant equations

f(x)=λe^(-λx)
f(y)=λe^(-λy)
f(x,y) = (λ^2)e^(-λ(x+y))
Kurtosis = E(((W-μ)/σ)^4)-3

3. The attempt at a solution
I first attempted to find the cdf of W, broken up into two parts: -∞<w<0 and 0<w<∞.
-∞<w<0 : ∫from -w to ∞ ∫from 0 to y+w (f(x,y))dxdy = (1/2)e^(λw)
0<w<∞ : ∫from 0 to ∞ ∫from 0 to y+w (f(x,y))dxdy = -(1/2)e^(-λw) + 1

I then derived the cdf to get a pdf of
f(w) = (λ/2)e^(λw) , -∞<w<0
(λ/2)e^(-λw) , 0<w<∞

For the coefficient of kurtosis, I kept the problem broken into the two regions of W.
For -∞<w<0:
μ= -1/2λ and σ= √(3)/2λ
Kurtosis = ∫from -∞ to 0 (((2λw+1)^4)/9)*((λ/2)e^(λw))dw - 3
For 0<w<∞:
μ= 1/2λ and σ= √(3)/2λ
Kurtosis = ∫from 0 to ∞ (((2λw-1)^4)/9)*((λ/2)e^(-λw))dw - 3

However, when I tried to calculate in mathematica, the program could not complete it. I have no idea what's wrong or what part of the problem needs to be corrected, but if someone could please look at it and let me know, that would be wonderful!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Dec 19, 2012

Ray Vickson

The Kurtosis is $E(W - EW)^4/\sigma^4 \;- \; 3$, and the density function f(w) of W is symmetric about w = 0, so EW = 0!

It is a very bad mistake to compute means of the separate (w > 0) and (w < 0) parts; W just has one part, and it goes from -∞ to +∞. The density function f(w) has a different formula in the two regions {w>0} and {w<0}, but that is a separate issue!