Statistics 270 - Lecture 12

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

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

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

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

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

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?

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

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

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

CDF for the Continuous Uniform

Example CDF Suppose that X has pdf: cdf:

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