Use the series expansion method of the MGF to find E(X), E(X2) and E(X3), the first three moments of X about the origin. Use your results to show that the mean and variance of X are both equal to 2.
The standardised third moment, ?3 = E{ (X – ?)3 ÷ ?3 }, provides a measure of skewness in the distribution of X. Determine the value of ?3 for this distribution.
Using a suitable distributional approximation, find the probability that the mean of a random sample of n = 25 observations will exceed 2.5.
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Question (Computer output is NOT admissible) The continuous random variable X has probability density function (pdf) given by Show that, subject to t <><>
Question (Computer output is NOT admissible) The continuous random variable X has probability density function (pdf) given by otherwise X e x x f X 0 0 ) ( = ³ * = - Show that, subject to t < 1, the moment generator function, mgf, of x is: 2 ) 1 ( 1 ) ( t t m x - = use the series expansion method of the mgf to find e(x), e(x2) and e(x3), the first three moments of x about the origin. use your results to show that the mean and variance of x are both equal to 2. the standardised third moment, 3 = e{ (x – )3 ÷ 3 }, provides a measure of skewness in the distribution of x. determine the value of 3 for this distribution. using a suitable distributional approximation, find the probability that the mean of a random sample of n = 25 observations will exceed 2.5. _121124552.unknown _121124232.unknown 1,="" the="" moment="" generator="" function,="" mgf,="" of="" x="" is:="" 2="" )="" 1="" (="" 1="" )="" (="" t="" t="" m="" x="" -="Use" the="" series="" expansion="" method="" of="" the="" mgf="" to="" find="" e(x),="" e(x2)="" and="" e(x3),="" the="" first="" three="" moments="" of="" x="" about="" the="" origin.="" use="" your="" results="" to="" show="" that="" the="" mean="" and="" variance="" of="" x="" are="" both="" equal="" to="" 2.="" the="" standardised="" third="" moment,="" 3="E{" (x="" –="" )3="" ÷="" 3="" },="" provides="" a="" measure="" of="" skewness="" in="" the="" distribution="" of="" x.="" determine="" the="" value="" of="" 3="" for="" this="" distribution.="" using="" a="" suitable="" distributional="" approximation,="" find="" the="" probability="" that="" the="" mean="" of="" a="" random="" sample="" of="" n="25" observations="" will="" exceed="" 2.5.="" _121124552.unknown="">