Thread: sample mean question

Y_1,Y_2, . . . ,Y_n are independent and identically distributed Exp(Q)
random variables.
Which are the distribution of the sample mean?

Sample mean=X=(1/n)*SIGMA(Y_k)

1. X~Exp(Q)
2. X~Exp(nQ)
3. X~Exp(Q/n)
4. X~Gamma(n,Q)
5. X~Gamma(n,(Q/n))

Originally Posted by MC Squidge

Y_1,Y_2, . . . ,Y_n are independent and identically distributed Exp(Q)
random variables.
Which are the distribution of the sample mean?

Sample mean=X=(1/n)*SIGMA(Y_k)

1. X~Exp(Q)
2. X~Exp(nQ)
3. X~Exp(Q/n)
4. X~Gamma(n,Q)
5. X~Gamma(n,(Q/n))

So, what attempts have you made?

the sum is distributed as by using MGFs
Next make a calc one change of variable to divide that rv by n.

I`m sorry can you do one of them as an example I don`t quite understand.

let

Then by using MGFs

So, and

SO, start by writing the density of W, which there are two ways, that's one reason I didn't
write my gamma density.

Ok thanks a lot for your help I think I`m beggining to understand it now. So 1,2,3 and 5 AREN'T distributions of the sample mean because their MGFs are different, but 4 is because its MGF is the same?

edit; actually it`s 5 that has the same MGF as the sample mean right? not 4

edit 2; yes i`ve definitely figured it out it`s 5 only. tell me if I`ve got it. thanks a lot for the help