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Chapter 4: Q6E (page 207)

Suppose that X and Y are independent random variableswhose variances exist and such that \(E\left( X \right) = E\left( Y \right)\).Show that\(E\left( {{{\left( {X - Y} \right)}^2}} \right) = Var\left( X \right) + Var\left( Y \right)\).

Short Answer

Expert verified

\(E\left( {{{\left( {X - Y} \right)}^2}} \right) = Var\left( X \right) + Var\left( Y \right)\).

Step by step solution

01

Given information

X and Y are independent random variables whose variances exist, and \(E\left( X \right) = E\left( Y \right)\).

02

Prove \(E\left( {{{\left( {X - Y} \right)}^2}} \right) = Var\left( X \right) + Var\left( Y \right)\)

Since X and Y are independent random variables, \(E\left( {XY} \right) = E\left( X \right)E\left( Y \right)\).

Also, \(\begin{aligned}{}Var\left( X \right)& = E\left( {{X^2}} \right) - {\left\{ {E\left( X \right)} \right\}^2}\\ \Rightarrow E\left( {{X^2}} \right) &= Var\left( X \right) + {\left\{ {E\left( X \right)} \right\}^2}\end{aligned}\).

Now,

\(\begin{aligned}{}E\left( {{{\left( {X - Y} \right)}^2}} \right) = E\left( {{X^2} - 2XY + {Y^2}} \right)\\ = E\left( {{X^2}} \right) - 2E\left( X \right)E\left( Y \right) + E\left( {{Y^2}} \right)\\ = Var\left( X \right) + {\left\{ {E\left( X \right)} \right\}^2} - 2E\left( X \right)E\left( Y \right) + Var\left( Y \right) + {\left\{ {E\left( X \right)} \right\}^2}\\ = Var\left( X \right) + Var\left( Y \right) + \underbrace {{{\left\{ {E\left( X \right)} \right\}}^2} - 2{{\left\{ {E\left( X \right)} \right\}}^2} + {{\left\{ {E\left( X \right)} \right\}}^2}}_{ = 0}{\rm{ }}\left( {{\rm{since, }}E\left( X \right) = E\left( Y \right)} \right)\\ = Var\left( X \right) + Var\left( Y \right)\end{aligned}\)

Hence, \(E\left( {{{\left( {X - Y} \right)}^2}} \right) = Var\left( X \right) + Var\left( Y \right)\).

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Suppose that the random variable X has the uniform distribution on the interval [0, 1], that the random variable Y has the uniform distribution on the interval [5, 9], and that X and Y are independent. Suppose also that a rectangle is to be constructed for which the lengths of two adjacent sides are X and Y. Determine the expected value of the area of the rectangle.

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Suppose that an observed value of X is equally likely to come from a continuous distribution for which the pdf is for from one for which the pdf is g. Suppose that \(f\left( x \right) > 0\) for \(0 < x < 1\) and \(f\left( x \right) = 0\) otherwise, and suppose also that \(g\left( x \right) > 0\) for \(2 < x < 4\) and \(g\left( x \right) = 0\) otherwise. Determine:

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