This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.

Exercises

• Let $Z ∼ N(0, 1)$, and let $X = 𝜎Z + 𝜇$ where $\mu$ and $\sigma > 0$ are constants. Let $\Phi$ represent the cumulative distribution function of $Z$, and let $\phi$ represent the probability density function, so $\phi(x) = \frac {1} {\sqrt 2 \pi }e^{\frac {−x^2}{2}}$.

  a. Show that the cumulative distribution function of $X$ is $F_X(x) = \Phi(x − \mu \sigma)$.

  b. Differentiate $F_X(x)$ to show that $X ∼ N(\mu, \sigma^2)$.

  c. Now let $X = −\sigma Z + \mu$. Compute the cumulative distribution function of $X$ in terms of $ \Phi$, then differentiate it to show that $X ∼ N(\mu, \sigma^2)$.