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If the random variable $X$ has the mean $\mu$ and the standard deviation $\sigma$, show that the random variable $Z$ whose values are related to those of $X$ by means of the equation

$$Z = \frac{X−\mu}{\sigma}$$ has $\mathbb{E}[Z] = 0$ and $\mathbb{Var}(Z) = 1$.

Does anyone have a solution for this problem? Thanks!

gt6989b
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Sparkles the unicorn
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1 Answers1

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HINT

You mean $Z = (X - \mu)/\sigma$, not $X - \mu/\sigma$. Let's compute the expected value together. We have $$ \mathbb{E}[Z] = \mathbb{E}\left[\frac{X-\mu}{\sigma}\right] = \frac{\mathbb{E}[X - \mu]}{\sigma} = \frac{\mathbb{E}[X] - \mu}{\sigma} = \frac{\mu - \mu}{\sigma} = 0. $$ Can you compute $\mathbb{E}\left[Z^2\right]$ and the resulting variance?

gt6989b
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  • How come you are able to pull out the $\sigma$ here and not take the expected value of it? (in your third step) @gt6989b – Sparkles the unicorn Feb 13 '20 at 04:39
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    @Sparklestheunicorn Expected value is linear. One way to see this is from the definition, i.e. if $X$ is continuous with pdf $f$ you have $$\mathbb{E}\left[\frac{X-\mu}{\sigma}\right] = \int_\mathbb{R} \frac{x-\mu}{\sigma} f(x) dx = \frac{1}{\sigma} \int_\mathbb{R} (x-\mu) f(x)dx = \frac{\mathbb{E}[X-\mu]}{\sigma}$$ – gt6989b Feb 13 '20 at 04:46