Question

DISCUSS ANY ONE CONTINUOUS PROBABILITY DISTRIBUTION?

Answer 1

Question #12776

DISCUSS ANY ONE CONTINUOUS PROBABILITY DISTRIBUTION.

Solution.

Here we will discuss normal distribution $N(\mu, \sigma^2)$. The distribution density function is $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$, $x \in (-\infty, +\infty)$. Mean equals $\mu$, variance equals $\sigma^2$. Characteristic function equals $\varphi(t) = e^{i\mu t - 1/2t^2\sigma^2}$. If $X \sim N(\mu, \sigma^2)$ then $\frac{X - \mu}{\sigma}$ has standard normal distribution $N(0,1)$. The normal random $X$ variable with 'big' probability takes values near $\mu$. For instance, $P(|X - \mu| \geq 3\sigma) \approx 0.0027$.