Question

In Example 6.33 we found the distribution of the sum of two i.i.d. exponential variables with parameter λ. Call the sum X. Let Y be a third independent exponential variable with parameter λ. Use the convolution formula 6.8 to find the sum of three independent exponential random variables by finding the distribution of X + Y.

Answer 1

Answer:

Given Data

we found the distribution of the sum of two i.i.d. exponential variables with parameter λ.

Let Y be a third independent exponential variable with parameter λ.

x = sum of two expoential distribution.

distribution

and   distribution

let pdf of x be f(x)

and pdf of y be g(y)

need to find pdf of z = x + y

so let pdf of z be h(z)

then by convolution formula:

now

  

  

,   ,

,

  

     

so

  

Hence

,  

so distribution

same way it can be proven that distribution

if each distribution.