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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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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
up vote 2 down vote accepted

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

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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Upvoted! $ $ $ $ – Did Aug 12 '12 at 14:03
@did Ahah! And finally I knew the secret of posting strings that satisfy strlen(string) < 15! – Frenzy Li Aug 12 '12 at 14:05
I like your answer, but I think letting the first link be a (very) comprehensive article about the normal distribution, from which he only needs to extract what the mean and variance of a normal distribution, might be a bit discouraging. (Given the difficulty of the question). – Henrik Aug 12 '12 at 15:55
@Henrik: Sorry but your conjecture is now disproved, see which answer got accepted. – Did Aug 12 '12 at 17:34
@FrenzY Thanks for your answer. I found the formulas that were given above in the articles. Although was a bit confusing! – Fred Aug 12 '12 at 18:33

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