### Q Marcum Function

## Gussian Random Variable

### Normal Distribution

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

This applet computes probabilities and percentiles for normal random variables: $$X \sim N(\mu, \sigma)$$

#### Directions

- Enter the mean $\mu$ and standard deviation $\sigma$.
- To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue box and press "Tab" or "Enter" on your keyboard. The probability $P(X \lt x)$ will appear in the pink box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability.
- To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink box, select $P(X \lt x)$ from the drop-down box, and press "Tab" or "Enter" on your keyboard. The percentile $x$ will appear in the blue box.

On the graph, the $x$ value appears in blue while the probability is shaded in pink.

#### Details

- Probability density function $$f(x)=\frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{1}{2\sigma^2}(x-\mu)^2}$$ where $-\infty \lt x \lt \infty$, $-\infty \lt \mu \lt \infty$, and $\sigma \gt 0$
- $\mu=E(X)$
- $\sigma^2=Var(X)$
- $\sigma=SD(X)$