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Probability with Applications in Engineering, Science, and Technology

Suppose $X \sim N(\mu, \sigma) .$

(a) Show via integration that $E(X)=\mu .[$Hint: Make the substitution $u=(x-\mu) / \sigma,$ which will create two integrals. For one, use the symmetry of the pdf; for the other, use the fact that the standard normal pdf integrates to $1 . ]$

(b) Show via integration that $\operatorname{Var}(X)=\sigma^{2}.$ [Hint: Evaluate the integral for $E\left[(X-\mu)^{2}\right]$ rather than using the variance shortcut formula. Use the same substitution as in part (a).]