Given that X and Y are independent and normally distributed as N(0,3) and N(0,5) respectively, what is the expected value of (XY)^2?

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    $\begingroup$ Are $X$ and $Y$ independent? $\endgroup$
    – Henry
    Jan 24 at 11:52
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    $\begingroup$ @Henry yes, they are $\endgroup$ Jan 24 at 12:16

1 Answer 1


The trick here is to rewrite the expected value in terms of several rules that you have learned in class. Maybe the following rules ring a bell:

  • $E[XY] = E[X]\cdot E[Y] \quad \text{if $X$ and $Y$ are independent}$
  • $E[X^2] = \text{Var}[X]+ E[X]^2$
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    $\begingroup$ One more rule is needed stats.stackexchange.com/questions/52646/… $\endgroup$ Jan 24 at 14:15
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    $\begingroup$ @Jacques That extra rule is not needed and applying it only complicates the solution. Another one that is used implicitly is that functions of independent variables (separately) remain independent. For this specific question I would point out that $E[X^2]=\operatorname{Var}(X)$ because $E[X]=0.$ That eliminates all probability calculations and allows one to write down the answer immediately, depending on how "N(0,3)" is interpreted (is its variance $3$ or $9$?). (+1) $\endgroup$
    – whuber
    Jan 24 at 15:10
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    $\begingroup$ @JacquesWainer: The more useful rule is that $(XY)^2 = X^2Y^2$. $\endgroup$ Jan 24 at 19:53

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