# moment-generating function

• May 3rd 2008, 03:56 PM
jorgeston
moment-generating function
\$\displaystyle Y\$ have pdf \$\displaystyle P(Y=k)=(k+1)p^2 q^k\$, with \$\displaystyle 0<p<1 \$and \$\displaystyle q=1-p\$ and \$\displaystyle k=0,1,2,...\$
Is easy prove that \$\displaystyle M_{Y}(t)=p^2 (1-qe^t)^2\$. Let \$\displaystyle X=Y_1+\cdots Y_{24}\$, with \$\displaystyle Y_{i}\$ with same distribution of \$\displaystyle Y\$
Calculate \$\displaystyle M_{X}(t)\$

Note: \$\displaystyle M_Y (t)=E(e^{tk})\$
• May 3rd 2008, 04:08 PM
mr fantastic
Quote:

Originally Posted by jorgeston
\$\displaystyle Y\$ have pdf \$\displaystyle P(Y=k)=(k+1)p^2 p^k\$, with \$\displaystyle 0<p<1 \$and \$\displaystyle q=1-p\$ and \$\displaystyle k=0,1,2,...\$
Is easy prove that \$\displaystyle M_{Y}(t)=p^2 (1-qe^t)^2\$. Let \$\displaystyle X=Y_1+\cdots Y_{24}\$, with \$\displaystyle Y_{i}\$ with same distribution of \$\displaystyle Y\$
Calculate \$\displaystyle M_{X}(t)\$

Note: \$\displaystyle M_Y (t)=E(e^{tk})\$

I have no time now. I'll reply later if no-one else does.
• May 3rd 2008, 04:46 PM
tukeywilliams
\$\displaystyle X = 24Y \$. Note that \$\displaystyle M_{aY+b} = e^{bt} M_{Y}(at) \$. In this case, \$\displaystyle b = 0 \$ and \$\displaystyle a = 24 \$. So \$\displaystyle M_{24Y}(t) = M_{X}(t) = p^2(1-qe^{24t})^{2} \$.
• May 3rd 2008, 06:43 PM
jorgeston
Quote:

Originally Posted by tukeywilliams
\$\displaystyle X = 24Y \$. Note that \$\displaystyle M_{aY+b} = e^{bt} M_{Y}(at) \$. In this case, \$\displaystyle b = 0 \$ and \$\displaystyle a = 24 \$. So \$\displaystyle M_{24Y}(t) = M_{X}(t) = p^2(1-qe^{24t})^{2} \$.

If you have reason, i don't understand why \$\displaystyle Y_1=Y_2=\cdots = Y_{24}\$

If this is true, implies \$\displaystyle X=24Y\$, but i am confused
• May 3rd 2008, 07:34 PM
mr fantastic
Quote:

Originally Posted by jorgeston
If you have reason, i don't understand why \$\displaystyle Y_1=Y_2=\cdots = Y_{24}\$

If this is true, implies \$\displaystyle X=24Y\$, but i am confused

*Ahem* Didn't you say in your question:

"with http://www.mathhelpforum.com/math-he...5618adbc-1.gif with same distribution of http://www.mathhelpforum.com/math-he...d003a3e0-1.gif" ....?