Today's Topic: The (Continuous) Normal Distribution
- Understand that there are infinitely many continuous probability distributions such as the Uniform, Normal, and Chi-squared distributions
- Understand that the normal curve is a parameterized function of x with parameters mu and sigma (mean and standard deviation)
- Use the Empirical Rule to calculate probabilities under the normal curve
- Calculate z-scores and use them to compare quantities
Non-uniform Probability Density Functions
- Area under the curve always equals one (for 100%)
- Can be any shape as long as the above holds true
- Not common
The Uniform Distribution
- Also called the rectangular distribution
- It has a constant probability over its interval
- Mean = average of max and min
- To find probabilities, just find the area of the rectangle defined by the limits given
The Gaussian Distribution (ie. The Standard Normal Curve)
- Perfectly Symmetric
- Centered at a mean of zero
- Standard Deviation of one
- Can be shifted and scaled by the parameters in order to model various scenarios that are approximately normal
The Empirical Rule
Input y = (normal curve function) using a mean of 0 and standard deviation of 1.
- Set your window to x: [-3, 3] y: [0, 0.5]
- Integrate between +1 SD, +2 SD, and +3 SD to find the probability of landing in each region
Change the SD to 2.
- Integrate between each set of standard deviations to find the probability of landing in each region
Isn't there an easier way?
NormalPDF(x, m, s)!!!
A Z-score measures how many standard deviations an observation is from the mean. It is the standardized measure of the score.
Easiest formula in this class: Z = (x - m)/s