STAT 462 Homework 3 due Tuesday, September 11, 2012 1) An appliance store receives a shipment of 30 microwave ovens, 5 of which are (unknown to the manager) defective. The store manager selects 4...

STAT 462
Homework 3
due Tuesday, September 11, 2012
1) An appliance store receives a shipment of 30 microwave ovens, 5 of which are (unknown to the
manager) defective. The store manager selects 4 ovens at random, without replacement, and tests
to see if they are defective. Let X = number of defectives found. Calculate the pmf and cdf of X
and plot the cdf.
2) A certain river floods every year. Suppose that the low-water mark is set at 1 and the high-water mark
Y has distribution function
2
0 1
( )= ( )= . 1 1 1
? ? ? ? ? ? ? ? ?
?
Y
y
F y PY y
y
y
a) Verify that F y Y ? ? is a cdf.
b) If the low-water mark is reset at 0 and we use a unit of measurement that is 1
10 of that given
previously, the high-water mark becomes Z Y = 10( 1) ? . Find ( ) F z Z .
3) A popular distribution for modeling waiting times is the exponential distribution. Let X be the
amount of time (in minutes) a randomly chosen person waits in line at a movie theatre. The random
variable X follows an exponential distribution where, for ? > 0, the CDF of X is
0 0
( )= ( )= 1 0 ??
? ?
? ?
? ? ? X x
x
F x PX x
e x
a) Using the CDF of X , find the probability a person waits more than 1
t minutes.
b) Given a person has already been waiting at least 0t minutes, find the probability the person
waits an additional 1
t minutes or more. How does this probability compare to your answer in
part (a)?
c) The answers to parts (a) and (b) illustrate the memoryless property of exponential distributions.
Based on your answers to parts (a) and (b), give a definition for the term memoryless property.