Statistics 270 Lecture 12 Last dayToday More discrete

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, 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: