Question

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 ?.

Answer 1

If
`X,Y` are independent, we have the properties for expectation and variance,

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

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

where
`a,b\in\Bbb R` are fixed.

Then

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

and

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

(recall that standard deviation = √(variance))