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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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    @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.2012-08-12

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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! :)2012-08-12
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Not an answer:

Check out Wikipedia, and then learn them through comprehension and by heart.

  1. Normal Distribution (E, $\sigma$ included)

  2. What is Variance

  3. Important Properties of Variance

  4. Important Properties of Expected Value

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    @FrenzYDT. You're fine! As I said any help was much appreciated.2012-08-14