# Express Expectation and Variance in other terms.

Let $X \sim N(\mu,\sigma^2)$ and $$f_X(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$ where $-\infty < x < \infty$.

Express $\operatorname{E}(aX + b)$ and $\operatorname{Var}(aX +b)$ in terms of $\mu$, $\sigma$, $a$ and $b$, where $a$ and $b$ are real constants.

This is probably an easy question but I'm desperate at Probability! Any help is much appreciated as I'm not even sure where to start.

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If you're studying this, then surely you have access to formulas relating $E(aX+b)$ to $E(X)$? –  Gerry Myerson Aug 12 '12 at 12:52
@GerryMyerson I have seen those formulas before but I think I was more so thrown off at the question as it was going for the same amount of marks as trickier ones. Also I don't have many 'useful' notes in this subject. It seems ridiculously easy now. –  Fred Aug 12 '12 at 18:41

If $a,b$ are constants, i.e. not random, then $$\mathbb{E}(aX+b) = a\mathbb{E}(X)+b,$$ $$\operatorname{var}(aX+b) = a^2 \operatorname{var}(X).$$

Now plug in $\mu$ and $\sigma^2$ in the appropriate places.

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Oh wait so aμ + b, (a^2)(σ^2)? I'm being completely thrown off by these exam questions because some are difficult and some easy but worth the same amount of marks. Thanks so much anyway! :) –  Fred Aug 12 '12 at 17:29

Upvoted!   –  Did Aug 12 '12 at 14:03