How to calculate the expected value of a standard normal distribution?

I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is E[X]=xf(x)dx where f(x) is the probability density function of X.

Suppose the probability density function of X is f(x)=12πex22 which is the density of the standard normal distribution.

So, I would first plug in the PDF and get
E[X]=x12πex22dx
which is a rather messy looking equation. The constant 12π can be moved outside the integral, giving
E[X]=12πxex22dx.

I get stuck here. How do I calculate integral? Am I doing this correctly this far? Is the simplest way to get the expected value?

Answer

You are almost there,
follow your last step:

E[X]=12πxex22dx=12πex2/2d(x22)=12πex2/2=0.

Or you can directly use the fact that xex2/2 is an odd function and the limits of the integral are symmetry.

Attribution
Source : Link , Question Author : mmh , Answer Author : Deep North

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