ELEC 321

# Normal Distribution

Updated 2017-10-11

## Standard Normal

Standard Normal Random Variable is denoted by $Z$. The notation $Z$ ~ $N(0,1)$ means that “$Z$ is a normal random variable mean of 0 and variance of 1”.

### Density Function

The standard normal density is given by

### Distribution Function

The standard normal distribution function is given by

Note: $\Phi(z)$ cannot be calculated in close form

Therefore, it is usually better to use the standard normal table or the function pnorm(z).

Due to the symmetry of the distribution function

### Mean

The mean, or expected value is given by (as always):

Notice the expected value for standard normal is at 0 since the standard normal centers around 0.

### Variance

Example: concrete mix

A machine fills 10-pound bags of dry concrete mix. The actual weight of the mix put into the bag is a normal random variable with standard deviation $\sigma=0.1$ pound. The mean can be set by the machine operator

a. is the mean at which the machine should be set if at most 10% of the bags can be underweight?

Let $X\sim \text{Norm}(\mu, \sigma^2)$ where $X$ is the actual weight. Thus we can express the following.

Which means the probability of weight less than 10 pounds is 0.1.

### Standard Deviation

Since the variance equals to 1, standard deviation also equals to 1: $\sigma=1$.

## Measurement Error Model

Suppose we have:

• Measurements $X_i$ ($X_1,X_2,\dotsc,X_n$)
• “True” value $\mu$
• “Inverse precision” of the measurements (variance) $\sigma$
• Measurement error in the standard scale $Z_i\sim \text{Norm}(0,1)$
• Measurement error in the original scale $\sigma Z_i$

Then we can model the errors as follows.

Using this equation, we can find the error of the individual measurement to be

## General Normal Random Variables

This applies to any normal random variables that aren’t standardized. These random variables are denoted as $X\sim \text{Norm}(\mu,\sigma^2)$, which stands for “X is a normal random variable with a mean of $\mu$ and a variance of $\sigma^2$”.

Manipulating the mean ($\mu$) shifts the distribution left and right. Manipulating the variance ($\sigma^2$) changes the amplitude and thickness of the distribution.

### Mean and Variance

Recall that $X=\mu+\sigma Z$ and $Z\sim\text{Norm}(0,1)\iff Z=\frac{X-\mu}{\sigma}$ , we can substitute $Z$ into $X$ and find the expected value and variance functions.

### Distribution Function

First, start with the definition of distribution function.

Next, we subtract $\mu$ and divide $\sigma$ on both sides of the inner inequality.

Recall that $Z=\frac{X-\mu}{\sigma}$, we plug it in.

Notice that this is the standard normal distribution function. Thus,

### Density Function

Recall that $F'(X)=f(x)$and $\Phi'(z)=\varphi(z)=\frac{1}{\sqrt{2\pi}}e^{-\frac12z^2}$, the density function is simply as follows.

Example:

Let $Z\sim\text{Norm}(0,1)$, calculate:

• Note that $\Phi(x)$ can be calculated in R using the pnorm(x) function.

• Find such that $\mathbb P(Z\gt c)=0.05$

Note that the inverse of standard normal CDF function can be calculated in R using qnorm(0.95)

• Find $c$ such that $% $

Rearrange the terms we can find $\Phi(c)$. Once again, we can use the qnorm(c) function in R to find $c$.

Example:

Let $X\sim \text{Norm}(3, 25)$, calculate:

• $c$ such that $\mathbb P(X>c)=0.10$