Topic B6 — Continuous Distributions
Table of contents
Probability density function
The area under the function over all values of equals 1. That is, the integral equals 1.
No probability attached to a specific value.
Cumulative distribution function (CDF)
Probabilty that random variable is below . S-shaped function of x.
Proba that random var is between a and b
Uniform distribution
All the values have the same probability of happening. Computed as (height of the fnct) as the proba of , 0 for anything else.
Compute expected value
The middle of the range,
Variance :
Compute CDF
Uniform distrib of tips between 50 and 125.
Height?
Proba of having between max and 100?
Normal distribution
$f(x)= \frac{1}{\sqrt{2\pi \sigma^2}} exp \left{ \frac{(x-\mu)^2}{2\sigma^2} \right} $ don’t need to know this by heart!
It is completely described by and
Symmetry around the mean
There is 50% probability that and
Changes in vs changes in
Change in : moves on the x-axis
Changes in : bigger –> lower crest, thicker tails.
Standard normal distribution
Normal with mean = 0, std dev = 1
Standardization of normal random variables
If we got a normally distributed variable, we can standardize it using this formula . That is, apply this formula to all the observations , also called z-scores. Std Dev becomes 1 (as seen above)
Inverse transformation, simple algebra. You want isolated on the left of the = sign.
Tables help you and give you . Look for the first decimal down the first row, and second decimal across the columns
The probability for negative values: