1. Statistics Lecture 12

2. Last day/Today: More discrete probability distributions
Assignment 4: Chapter 3: 5, 7,17, 25, 27, 31, 33, 37, 39, 41, 45, 47, 51, 65, 67, 77, 79

3. Continuous Random Variables
For discrete random variables, can assign probabilities to each outcome in the sample space
Continuous random variables take on all possible values in an interval(s)
Random variables such as heights, weights, times, and measurement error can all assume an infinite number of values
Need different way to describe probability in this setting

4. Can describe overall shape of distribution with a mathematical model called a density function, f(x)
Describes main features of a distribution with a single expression
Total area under curve is
Area under a density curve for a given range gives

7. Use the probability density function (pdf), f(x), as a mathematcal model for describing the probability associated with intervals
Area under the pdf assigns probability to intervals

8. Example
A college professor never finishes his lecture before the assigned time to end the period
He always finishes his lecture within one minute assigned end of class
Let X = the time that elapses between the assigned end of class and the end of the actual lecture
Suppose the pdf for X is

9. Example What is the value of k so that this is a pdf?
What is the probability that the period ends within ½ minute of the scheduled end of lecture?

10. Example (Continuous Uniform)
Consider the following curve:
Draw curve:
Is this a density?

11. Example (Continuous Uniform)
In general, the pdf of a continuous uniform rv is:
Is this a pdf?

13. CDF
Recall the cdf for a discrete rv
The cdf for the continuous rv is:

14. CDF for the Continuous Uniform

15. Example CDF
Suppose that X has pdf:
cdf:

16. Using the CDF to Compute Probabilities
Can use cdf to compute the probabilities of intervals…integration
Can also use cdf: