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

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- 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

intervalsintegration - Can also use cdf