Let X N(0,1). Find Cov(X, X2).

Expand the covariance. By definition, Cov(X,X2) = E[X3] - E[X]E[X2]. Evaluate the moments. For a standard normal, all odd moments vanish, so E[X3]=0 and E[X]=0.

Zero Covariance Without Independence

Statistics

Dependence

Medium

Let

X \sim N (0, 1)

. Find

Cov (X, X^{2})

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Hint 1

Write the covariance as $E [X^{3}] - E [X] E [X^{2}]$ .

Hint 2

Use the fact that all odd moments of a standard normal are zero.

Show worked solution

1. Expand the covariance
By definition, $Cov (X, X^{2}) = E [X^{3}] - E [X] E [X^{2}]$ .
2. Evaluate the moments
For a standard normal, all odd moments vanish, so $E [X^{3}] = 0$ and $E [X] = 0$ .
$Cov (X, X^{2}) = 0 - 0 \cdot E [X^{2}] = 0$

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Zero Covariance Without Independence

1. Expand the covariance

2. Evaluate the moments

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