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))
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))
let $\displaystyle W=Y_1+\cdots +Y_n$
Then by using MGFs $\displaystyle W\sim\Gamma(n,Q)$
So, $\displaystyle X=W/n$ and $\displaystyle f_X(x)=f_W(w)\bigl|{dw\over dx}\bigr|$
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