# Derive the moment-generating function of Y bar

• Nov 25th 2011, 08:06 PM
wopashui
Derive the moment-generating function of Y bar
Suppose that Y1, Y2, …, Yn are independent, normally distributed random variables with mean u and
variance $\displaystyle \sigma^2$ . Define
$\displaystyle Y bar= \displaystyle \sum^n_{i=1} Yi/n$

Derive the moment-generating function of Y bar
• Nov 25th 2011, 08:29 PM
mr fantastic
Re: Derive the moment-generating function of Y bar
Quote:

Originally Posted by wopashui
Suppose that Y1, Y2, …, Yn are independent, normally distributed random variables with mean u and
variance $\displaystyle \sigma^2$ . Define
$\displaystyle Y bar= \displaystyle \sum^n_{i=1} Yi/n$

Derive the moment-generating function of Y bar

What have you tried? Where are you stuck?
• Nov 26th 2011, 08:38 AM
wopashui
Re: Derive the moment-generating function of Y bar
Quote:

Originally Posted by mr fantastic
What have you tried? Where are you stuck?

srooy, i don't know how to start the question, do i start by definition m(t)= E(e^tybar)= integral e^tybar times f(y) then sub in the summation into the integral, but what is f(y)here, do we use the normal density function?
• Nov 26th 2011, 11:32 AM
mr fantastic
Re: Derive the moment-generating function of Y bar
Quote:

Originally Posted by wopashui
srooy, i don't know how to start the question, do i start by definition m(t)= E(e^tybar)= integral e^tybar times f(y) then sub in the summation into the integral, but what is f(y)here, do we use the normal density function?

Given!!:

Quote:

Yn are independent, normally distributed random variables with mean u and
variance img.top {vertical-align:15%;}http://latex.codecogs.com/png.latex?\sigma^2 .
• Nov 30th 2011, 05:07 PM
matheagle
Re: Derive the moment-generating function of Y bar
well linear combo's of normals is a normal, hence just compute moo and sigma squared of y bar.
• Dec 1st 2011, 08:16 AM
wopashui
Re: Derive the moment-generating function of Y bar
Quote:

Originally Posted by matheagle
well linear combo's of normals is a normal, hence just compute moo and sigma squared of y bar.

then how do you find the mgf with moo and the variance, we know that m'(0)=moo, and m''(0)-[m'(0)]^2= variance