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The distribution of random variable r has mean 10 and standard deviation 4. The distribution of random variable s has mean 7 and standard deviation 3. If r and s are independent, what are the mean and standard deviation of the distribution of r−s ?.

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LammettHash

If [tex]X,Y[/tex] are independent, we have the properties for expectation and variance,

[tex]\Bbb E[aX + bY] = a\,\Bbb E[X] + b\,\Bbb E[Y][/tex]

[tex]\Bbb V[aX + bY] = a^2\,\Bbb V[X] + b^2\,\Bbb V[Y][/tex]

where [tex]a,b\in\Bbb R[/tex] are fixed.

Then

[tex]\Bbb E[R - S] = \Bbb E[R] - \Bbb E[S] = 10 - 7 = \boxed{3}[/tex]

and

[tex]\Bbb V[R - S] = \Bbb V[R] + (-1)^2\, \Bbb V[S] = 4^2 + 3^2 = \boxed{5}\,{}^2 = 25[/tex]

(recall that standard deviation = √(variance))

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