Question

# Consider a random variable X following the exponential distribution X ~ f(x), where f(x) = ae^(...

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.

The pdf of X is The MGF will be So MGF is -------------------

Differentiating MGF 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 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 #### Earn Coins

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