ELEC 321
# Normal Distribution

Updated 2017-10-11

*Standard Normal Random Variable* is denoted by . The notation ~ means that “ is a normal random variable mean of 0 and variance of 1”.

The standard normal density is given by

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 mixA 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.

Since the variance equals to 1, standard deviation also equals to 1: .

Suppose we have:

- Measurements ()
- “True” value
- “Inverse precision” of the measurements (variance)
- Measurement error in the
*standard*scale - Measurement error in the original scale

Then we can model the errors as follows.

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

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.

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

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,

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

Example:Let , calculate:

Notethat can be calculated inRusing the`pnorm(x)`

function.Find such that

Notethat the inverse of standard normalCDFfunction can be calculated inRusing`qnorm(0.95)`

Find such that

Rearrange the terms we can find . Once again, we can use the

`qnorm(c)`

function inRto find .

Example:Let , calculate:

such that