Normal Distribution

The MOST powerful distribution. Normal distribution is a probability distribution that is symmetric about the mean , where data near the mean occur more frequently.

Normal Distribution (Definition)

We say that a r.v. is normally distributed, , if its p.d.f., , is


  • is the mean or expectation of the distribution (and also its median and mode),
  • is its Standard Deviation

Normal Distribution is symmetrical around , i.e. where is the mean, mode, and median.

Expectation and Variance

  • (the calculation for this is pretty messy)

Standard Normal Distribution (Definition)

The standard Normal Distribution has and , i.e. . We have the p.d.f.

I learned the matrix form from the SLAM book.

Multi-dimensional Normal Distribution (Matrix-Form Definition)

The probability density function is

Also see Gaussian Variable.

Where do we see gaussians in the real world?

  • Height Distribution: Human heights follow a Gaussian distribution.
  • IQ Scores: IQ scores typically follow a Gaussian distribution.
  • Measurement Errors: Instrument measurement errors often have a Gaussian distribution.
  • Stock Returns: Daily stock returns frequently exhibit a Gaussian distribution.

Z-table (Standard Normal z-table)

The z-table table gives values of the CDF for and .

What is really cool is that we can go from the standard normal distribution to another normal distribution with a particular mean and variance, using the theorem below:


If and , then

Look at the page on Z-Score

Example Exercise

Example: Suppose that final grades in this course follow a normal
distribution with and .

  1. Find the probability that a student has a final grade of 83 or more.

  2. Calculate the 95th percentile.

  3. We have that , so find

  4. We solve this by using the Quantile table. Therefore, we conclude that

Part 2: A sample of 9 students is taken. Find

I’m not sure if I understand correctly, but there is this Variance of the mean value that I need to use, instead of the direct variance.

  1. (I don’t know if is correct, might be a typo)

The 68-95-99 rule

The 68-95-99 rule is based on the mean and standard deviation. It says:

  • 68% of the population is within 1 standard deviation of the mean.
  • 95% of the population is within 2 standard deviation of the mean.
  • 99.7% of the population is within 3 standard deviation of the mean.

Generating Normal Distributions in Code

To randomly sample from the Normal Distribution, we have a few options (Learned from Ericsson):