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Category:Software companies of GermanyQ:

Expected Value of a Ranged random variable

Suppose $X$ is a continuous random variable with range $[a,b]$.
I am asked to find $E[X]$.
I know that the expected value of a continuous R.V. is defined as
$E[X] = \int_a^b xf_X(x)dx = \int_a^b x\frac{1}{b-a}\frac{1}{\sqrt{2\pi}}e^{ -(x-\mu)^2/2}dx$
Then my work so far has been
$E[X] = \int_a^b x\frac{1}{b-a}\frac{1}{\sqrt{2\pi}}e^{ -(x-\mu)^2/2}dx$
$= \int_a^b x\frac{1}{b-a}\frac{1}{\sqrt{2\pi}}\frac{e^{ -(x-\mu)^2/2}}{\frac{e^{ -(x-\mu)^2/2}}{\frac{e^{ -(x-\mu)^2/2}}{b-a}}}dx$
$= \int_a^b x\frac{1}{b-a}\frac{1}{\sqrt{2\pi}}\frac{e^{ -(x-\mu)^2/2}}{(b-a)^2}dx$
$= \frac{1}{(b-a)\sqrt{2\pi}}\int_a^bxe^{ -(x-\mu)^2/2}dx$
And now it’s getting tricky. I have no idea how to get the $\frac{1}{\sqrt{2\pi}}$ factor in the original equation. Am I doing anything wrong? Any advice?


By the law of the unconscious statistician, you can’t get the $\frac{1}{\sqrt{2 \pi}}$ out of that integrand, but just leave it there to

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