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1

Defining Probabilities: Random Variables

• Examples:– Out of 100 heart catheterization procedures performed at

a local hospital each year, the probability that more than five of them will result in complications is

__________

– Drywall anchors are sold in packs of 50 at the local hardware store. The probability that no more than 3 will be defective is

__________

– In general, ___________

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Discrete Random Variables

• Example:– Look back at problem 2.53, page 55. Assume someone

spends $75 to buy 3 envelopes. The sample space describing the presence of $10 bills (T) vs bills that are not $10 (N) is:

_____________________________

– The random variable associated with this situation, X, reflects the outcome of the choice and can take on the values:

_____________________________

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Discrete Probability Distributions

• The probability that there are no $10 in the group is

P(X = 0) = ___________________

The probability distribution associated with the number of $10 bills is given by:

x 0 1 2 3

P(X = x)

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Another Example

• Example 3.8, pg 80

P(X = 0) =

_____________________

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Discrete Probability Distributions

• The discrete probability distribution function (pdf) – f(x) = P(X = x) ≥ 0

– Σx f(x) = 1

• The cumulative distribution, F(x) – F(x) = P(X ≤ x) = Σt ≤ x f(t)

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Probability Distributions

• From our example, the probability that no more than 2 of the envelopes contain $10 bills is

P(X ≤ 2) = F(2) = _________________

• The probability that no fewer than 2 envelopes contain $10 bills is

P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - F(1) = ________________

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Another View

• The probability histogram

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3

x

f(x)

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Your Turn …

• The output from of the same type of circuit board from two assembly lines is mixed into one storage tray. In a tray of 10 circuit boards, 6 are from line A and 4 from line B. If the inspector chooses 2 boards from the tray, show the probability distribution function associated with the selected boards being from line A.

x P(x)

0

1

2

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Continuous Probability Distributions

• Examples:– The probability that the average daily temperature in

Georgia during the month of August falls between 90 and 95 degrees is

__________

– The probability that a given part will fail before 1000 hours of use is

__________

– In general, __________

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Understanding Continuous Distributions

• The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is

• The probability that a given part will fail before 1000 hours of use is

-5 -3 -1 1 3 5

0 5 10 15 20 25 30

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Continuous Probability Distributions

• The continuous probability density function (pdf) f(x) ≥ 0, for all x ∈ R

• The cumulative distribution, F(x)

1)( dxxf

b

a

dxxfbXaP )()(

x

dttfxXPxF )()()(

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Probability Distributions

• Example: Problem 3.7, pg. 88

x, 0 < x < 1

f(x) = 2-x, 1 ≤ x < 2

0, elsewhere

1st – what does the function look like?

a) P(X < 120) = ___________________

b) P(50 < X < 100) = ___________________

{

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Your turn

• Problem 3.14, pg. 89