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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 $E(X)$?2012-08-12
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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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    Upvoted! $ $ $ $2012-08-12
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    @did Ahah! And finally I knew the secret of posting strings that satisfy strlen(string) < 15!2012-08-12
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    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).2012-08-12
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    @Henrik: Sorry but your conjecture is now disproved, see which answer got *accepted*.2012-08-12
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    @FrenzY Thanks for your answer. I found the formulas that were given above in the articles. Although was a bit confusing!2012-08-12
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    @Henrik *she But yes was a little daunting trying to find the correct formulas in the articles although I did manage to find them.2012-08-12
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    @did Wow sorry I really wasn't trying to attack anyone - It's just that I can still remember when I couldn't figure out problems like this one and then the first link would have scared me off. I myself trying to help someone would easily have done the same.2012-08-12
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    @Panda Sorry of the first link scared you Oo... I just wanted to say that the boxes to the upper right of distributions are extremely useful.2012-08-12
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    @FrenzYDT. You're fine! As I said any help was much appreciated.2012-08-14