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MAT 1235
Calculus II
Section 8.5
Probability
http://myhome.spu.edu/lauw
HW



WebAssign 8.5
(6 problems, 65 min.)
Quiz: 8.2, 8.5
Preview

Provide a 30-minute snapshot of
probability theory and its relationship
with integration.
Preview



Provide a 30-minute snapshot of
probability theory and its relationship
with integration.
Engineering: MAT2200 (3)
Math major/minor: MAT 3360 (5) 
Random Variables

Variables related to random behaviors
Example 1
Y=outcome of rolling a die
=
X=lifetime of a Dell computer
=
Q: What is a fundamental difference
between X and Y?
Continuous Random Variables

Take range over an interval of real
numbers.
Probability…
of an event = the chance that the event will
happen
Example 2
P(Y=1)=1/6
The chance of getting “1” is ___________
P(3≤X≤4)
The chance that the Dell computer breaks
down____________________
Probability…
…of an event = the chance that the event
will happen
…is always between 0 and 1.
Example 3
P(Y=7)=
P(0 ≤ X<)=
Probability Density Function


Continuous random variable X
The pdf f(x) of X is defined as
b
P  a  X  b    f  x  dx
a

The prob. info is “encoded” into the pdf
Probability Density Function
Properties:
1. f  x   0 for all x

2.
 f  x  dx  1

Example 4
1

2
4 x  12 x if 0  x 
f  x  
2
0
Otherwise
(a) Show that f(x) is a pdf of some random
variable X.
Example 4
1

2
4 x  12 x if 0  x 
f  x  
2
0
Otherwise
(b) Let X be the lifetime of a type of battery
(in years). Find the probability that a
randomly selected sample battery will last
more than ¼ year.
13
 0.8125
16
Average Value of a pdf


 xf  x  dx

Also called 1. Mean of the pdf f(x)
2. Expected value X
Example 4
1

2
4 x  12 x if 0  x 
f  x  
2
0
Otherwise
(c) Let X be the lifetime of a type of battery
(in years). Find the average lifetime of
such type of batteries.


 xf  x  dx

17
 0.354 years
48
Exponential Distribution
0
f  t    ct
ce


if t  0
if t  0
Used to model waiting times, equipment
failure times. It have a parameter c.
The average value is 1/c. So, c =
Example 5
The customer service at AT&T has an
average waiting time of 2 minutes.
Assume we can use the exponential
distribution to model the waiting time. Find
the probability that customer will be served
within 5 minutes.
Example 5
Let T be the waiting time of a customer.
0
f  t    ct
ce
if t  0
if t  0
0.92
Remarks

If a random variable is not given, be sure
to define it.