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Joint Probability Distributions
• The probabilities associated with two things
both happening, e.g. …
– probability associated with the hardness and tensile
strength of an alloy
• f(h, t) = P(H = h, T = t)
– probability associated with the lives of 2 different
components in an electronic circuit
• f(a, b) = P(A = a, B = b)
– probability associated with the diameter of a mold and
the diameter of the part made by that mold
• f(d1, d2) = P(D1 = d1, D2 = d2)
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Example
• Statistics were collected on 100 people selected
at random who drove home after a college
football game, with the following results:
No
Accident
accident
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TOTAL
Drinking
7
23
30
Not
drinking
6
64
70
TOTAL
13
87
100
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Joint Probability Distribution
• The joint probability distribution or probability
mass function for this data:
Accident
No
accident
g(x)
Drinking
0.07
0.23
0.3
Not
drinking
0.06
0.64
0.70
h(y)
0.13
0.87
1
– The marginal distributions associated with drinking
and having an accident are calculated as:
_______________________________________
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Conditional Probability Distributions
• The conditional probability distribution of the
random variable Y given that X = x is
f ( x, y )
f (y | x ) 
g( x )
• For our example, the probability that a driver
who has been drinking will have an accident is
f(accident | drinking) = ___________________
(note: refer to page 95 of your textbook for the conditional probability that a
variable falls within a range.)
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Statistical Independence
• If f(x | y) doesn’t depend on y, then f(x | y) = g(x)
and
f(x, y) = g(x) h(y)
• X and Y are statistically independent if and only
if this holds true for all (x, y) within their range.
• For our example, are drinking and being in an
accident statistically independent? Why or why
not?
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Continuous Random Variables
• f(x, y) is the joint density function of the
continuous variables X and Y, and the
associated probability is
P[( x, y )  A   f ( x, y )dxdy
A
for any region A in the xy plane.
• The marginal distributions of X and Y alone are

g ( x )   f ( x, y )dy


and
h( y )   f ( x, y )dx

• See examples 3.15, 3.17, 3.19, 3.20, and 3.22
(starting on pg. 93)
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4.1 Mathematical Expectation
• Example: Repair costs for a particular machine
are represented by the following probability
distribution:
x
$50
200
350
P(X = x)
0.3
0.2
0.5
• What is the expected value of the repairs?
– That is, over time what do we expect repairs to cost on
average?
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Expected value
• μ = E(X)
– μ = mean of the probability distribution
• For discrete variables,
μ = E(X) = ∑ x f(x)
• So, for our example,
E(X) = ________________________
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Your turn …
• By investing in a particular stock, a person can
take a profit in a given year of $4000 with a
probability of 0.3 or take a loss of $1000 with a
probability of 0.7. What is the investor’s
expected gain on the stock?
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Expected value of Continuous Variables
• For continuous variables,
μ = E(X) = _______
• Example: Recall from last time, problem 3.7 (pg. 88)
f(x) =
{
x,
2-x,
0,
0<x<1
1≤x<2
elsewhere
(in hundreds of hours.)
What is the expected value of X?
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• E(X) = ∫ x f(x) dx
= ________________________
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Functions of Random Variables
• Example 4.4. Probability of X, the number of cars passing
through a car wash in one hour on a sunny Friday afternoon,
is given by
x
P(X = x)
4
5
1/12 1/12
6
7
8
9
1/4
1/4
1/6
1/6
Let g(X) = 2X -1 represent the amount of money paid to the
attendant by the manager. What can the attendant expect to
earn during this hour on any given sunny Friday afternoon?
E[g(X)] = Σ g(x) f(x) = ____________________
= _______________________________
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