My Math Forum Derive the moment-generating function of Y bar

 November 25th, 2011, 08:04 PM #1 Senior Member   Joined: Nov 2009 Posts: 169 Thanks: 0 Derive the moment-generating function of Y bar Suppose that Y1, Y2, …, Yn are independent, normally distributed random variables with mean u and variance $\sigma^2$ . Define $Y bar= \displaystyle \sum^n_{i=1} Yi/n$ Derive the moment-generating function of Y bar
November 26th, 2011, 01:04 PM   #2
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Re: Derive the moment-generating function of Y bar

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 Originally Posted by 450081592 Suppose that Y1, Y2, …, Yn are independent, normally distributed random variables with mean u and variance $\sigma^2$ . Define $Y bar= \displaystyle \sum^n_{i=1} Yi/n$ Derive the moment-generating function of Y bar
I prefer working with characteristic functions. The characteristic function for Yk is exp(i?t-{(?t)^2}/2). Therefore for Yk/n it is exp(i?t/n-{(?t/n)^2}/2). For Y bar it is exp(i?t-{(?t)^2}/(2n)).

You can get the moments by expanding the characteristic function in powers of t, or by recognizing that Ybar is normally distributed with mean ? and variance (?^2)/n.

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