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It is known that the expected value of a standard normal random variable X is 0. But what happens when a random variable is defined upon a random variable with a non-standard normal distribution? Must standardization be done in order to have a solution? Example:

Suppose $X$~$N(0,16)$. Define:

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Find E(Y). Any clue solving such problems would be welcomed.

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  • $\begingroup$ hint : any symmetry in the problem ? $\endgroup$
    – Thomas
    Dec 18, 2013 at 8:15
  • $\begingroup$ if you want to study the law of Y you can also be more "bruteforce" and write something with integrals. Can you try that ? We can tell you if you are correct or not/help you $\endgroup$
    – Thomas
    Dec 18, 2013 at 8:16

1 Answer 1

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In this case $E[Y] = \Pr(|X|\lt 4)$ so at some stage you are going to have to find values for the cumulative distribution function of a normal distribution.

<grumpy old man>When I was young, you looked these up in a table for a standard normal distribution, so you had to standardise the problem. Nowadays computers will do the calculations for you if you tell them the mean and standard deviation, so you do not.</grumpy old man>

So to get the answer in R:

pnorm(4, mean=0, sd=sqrt(16)) - pnorm(-4, mean=0, sd=sqrt(16))
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