Download - MAT 1235 Calculus II Section 8.5 Probability
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MAT 1235Calculus II
Section 8.5
Probability
http://myhome.spu.edu/lauw
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HW
WebAssign 8.5 (6 problems, 65 min.) Quiz: 8.2, 8.5
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Preview
Provide a 30-minute snapshot of probability theory and its relationship with integration.
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Preview
Provide a 30-minute snapshot of probability theory and its relationship with integration.
Engineering: MAT2200 (3) Math major/minor: MAT 3360 (5)
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Random Variables
Variables related to random behaviors
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Example 1
Y=outcome of rolling a die
=
X=lifetime of a Dell computer
=
Q: What is a fundamental difference between X and Y?
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Continuous Random Variables
Take range over an interval of real numbers.
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Probability…
of an event = the chance that the event will
happen
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Example 2
P(Y=1)=1/6The chance of getting “1” is ___________
P(3≤X≤4)The chance that the Dell computer breaks
down____________________
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Probability…
…of an event = the chance that the event
will happen
…is always between 0 and 1.
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Example 3
P(Y=7)=
P(0 ≤ X<)=
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Probability Density Function
Continuous random variable X The pdf f(x) of X is defined as
The prob. info is “encoded” into the pdf
b
a
P a b f xX x d
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Probability Density Function
Properties:
1. 0 for all
2. 1
f x x
f x dx
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Example 4
(a) Show that f(x) is a pdf of some random variable X.
2 1
4 12 if 02
0 Otherwise
x x xf x
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Example 4
(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.
2 1
4 12 if 02
0 Otherwise
x x xf x
130.8125
16
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Average Value of a pdf
Also called 1. Mean of the pdf f(x) 2. Expected value X
xf x dx
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Example 4
(c) Let X be the lifetime of a type of battery (in years). Find the average lifetime of such type of batteries.
2 1
4 12 if 02
0 Otherwise
x x xf x
xf x dx
17
0.354 years48
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Exponential Distribution
Used to model waiting times, equipment failure times. It have a parameter c.
The average value is 1/c. So, c =
0 if 0
if 0ct
tf t
ce t
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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.
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Example 5
Let T be the waiting time of a customer.
0 if 0
if 0ct
tf t
ce t
0.92
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Remarks
If a random variable is not given, be sure to define it.