Thread: Derive the moment-generating function of Y bar

1. 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

2. Re: Derive the moment-generating function of Y bar

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?

3. Re: Derive the moment-generating function of Y bar

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?

4. Re: Derive the moment-generating function of Y bar

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!!:

Yn are independent, normally distributed random variables with mean u and
variance img.top {vertical-align:15%;}$\sigma^2$ .

5. 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.

6. Re: Derive the moment-generating function of Y bar

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