Problem 7.1 This problem shows once more that a conditionally Gaussian random variable has excess kurtosis. Let us assume that X and 2 are two

Problem 7.1 This problem shows once more that a conditionally Gaussian random variable has excess kurtosis.

Let us assume that X and σ2 are two random variables and that X|σ2 ∼ N(0, σ2), i.e. that conditioned on the value of σ2, X is mean-zero normal with variance σ2. Prove that:

E{X4}

var{X}2 = 3



1 +

var{σ2}

E{σ2}2



proving the claim of excess kurtosis when σ2 is not deterministic.