ELEC 321

Normal Distribution

Updated 2017-10-11

Standard Normal

Standard Normal Random Variable is denoted by . The notation ~ means that “ 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: 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


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.


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 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 where 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: .

Measurement Error Model

Suppose we have:

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 , which stands for “X is a normal random variable with a mean of and a variance of ”.

Manipulating the mean () shifts the distribution left and right. Manipulating the variance () changes the amplitude and thickness of the distribution.

Mean and Variance

Recall that and , we can substitute into and find the expected value and variance functions.

Distribution Function

First, start with the definition of distribution function.

Next, we subtract and divide on both sides of the inner inequality.

Recall that , we plug it in.

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

Density Function

Recall that and , the density function is simply as follows.


Let , calculate:


Let , calculate: