Question

Consider a random variable X following the exponential distribution X ~ f(x), where f(x) = ae^( -ax) for x > 0 and 0 otherwise, a > 0. Derive its moment-generating function MX(t) and specify its domain (where it is defined or for what t does the integral exist). Use it to compute the first four non-central moments of X and then derive the general formula for the nth non-central moment for any positive integer n. Also, write down the expression for variance, skewness, and kurtosis.

Answer 1

The pdf of X is

The MGF will be

So MGF is

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Differentiating MGF with respect to t gives:

Putting t=0 gives

  

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Differentiating MGF again with respect to t gives:

Putting t=0 gives

  

Differentiating MGF again with respect to t gives:

Putting t=0 gives

  

Differentiating MGF again with respect to t gives:

Putting t=0 gives

The general expression is

The variance will be

The skewness is

The Kurtosis is