October 23, 2002

Continuous Random Variables

 

I.                    Definition. A random variable X is said to be continuous if its set of possible values is an entire interval of numbers. A probability model for X is given by assigning to a set of outcomes A the probability P(A) equal to the area above A and under a curve.

II.                 Definition. A probability distribution function or probability density function (pdf) of a continuous random variable X is a function f(x) such that for any two numbers a and b with a £b:                                                                                                      

III.               Exercise. Suppose that a person is supposed to take a bus that arrives every 10 minutes. The person arrives at a bus stop in a random fashion. Let X = waiting tine for the bus. Find the pdf of X.   Find P(1£X£3).

 

 

 

 

 

 

 

 

 

IV.              Definition. A continuous random variable X is said to have a uniform distribution on the interval [A,B] if the pdf of X is

V.                 Exercise. (#7) The time X (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution with A=25 and B=35.                      

A.     Write the pdf and sketch its graph

B.     What is the probability that preparation time exceeds 33 minutes?                                  

C.      What is the probability that preparation time is within 2 minutes of the mean time?(Hint: identify  the mean from the graph of f(x)).

D.     For any a such that 25<a<a+2<35, what is the probability that preparation time is between a and a+2 minutes?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VI.              Exercise (#10) A family of pdf's that has been used to approximate the distribution of income, city     population size, and size of firms is the Pareto family. The family has two parameters, k and q, both >0, and the pdf is

                   

A.     Sketch the graph of f(x;k,q).

B.     Verify that the total area under the graph equals 1.

C.     If the random variable X has the pdf f(x;k,q), for any fixed b>q obtain an expression for P(X£b).

D.     For q<a<b, obtain an expression for the probability P(a£X£b).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VII.            Cumulative distribution function

Definition. The cumulative distribution function F(x) for a continuous random variable X is defined for every number x by

VIII.         Example. Find the cumulative distribution function for a uniform distribution on [A,B].Draw a graph of cdf.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IX.               The expected or mean value of a continuous random variable X with pdf  f(x) is :          

X.                 If X is a continuous random variable with pdf f(x) and h(x) is any function of X, then      

   

XI.               The variance of a continuous random variable X with pdf f(x) and mean value m is

XII.            Exercise.   Let X have a uniform distribution on the interval [A,B]. Compute E(X),V(X), and sx.