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Practice Problem: compute the zero-th order moment of a Gaussian random variable
A random variable X has the following probability density function:
$ f_X (x) = \frac{1}{\sqrt{2\pi} 3 } e^{\frac{-(x-3)^2}{18}} . $
Compute the moment of order one of that random variable. In other words, compute
$ E \left( X^1 \right) . $
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Answer 1
The moment of n-th order moment is defined as: $ E[X^{n}]=\int_{-\infty }^{\infty }x^{n}f_{X}(x)dx $
Therefore,
$ E[X^{1}]=\int_{-\infty }^{\infty }xf_{X}(x)dx=\frac{1}{3\sqrt{2\Pi }}\int_{-\infty }^{\infty }xe^{-\frac{(x-3)^{2}}{18}}dx=\frac{1}{3\sqrt{2\Pi }}9\sqrt{2\Pi }=3 $
Answer 2
Write it here.
Answer 3
Write it here.
