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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:

AccidentNo

accidentTOTAL

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:

– The marginal distributions associated with drinking and having an accident are calculated as:

_______________________________________

AccidentNo

accidentg(x)

Drinking 0.07 0.23 0.3

Not drinking

0.06 0.64 0.70

h(y) 0.13 0.87 1

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Conditional Probability Distributions

• The conditional probability distribution of the random variable Y given that X = x is

• 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.)

)(

),()|(

xg

yxfxyf

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

for any region A in the xy plane.• The marginal distributions of X and Y alone are

• See examples 3.15, 3.17, 3.19, 3.20, and 3.22 (starting on pg. 93)

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A

dxdyyxfAyxP ),(),[(

dxyxfyhanddyyxfxg ),()(),()(

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4.1 Mathematical Expectation

• Example: Repair costs for a particular machine are represented by the following probability distribution:

• What is the expected value of the repairs?– That is, over time what do we expect repairs to cost on

average?

x $50 200 350

P(X = x) 0.3 0.2 0.5

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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)

x, 0 < x < 1

f(x) = 2-x, 1 ≤ x < 2

0, 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

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) = ____________________

= _______________________________

x 4 5 6 7 8 9

P(X = x) 1/12 1/12 1/4 1/4 1/6 1/6