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, ___________

Discrete Random Variables

• Example:– Look back at problem 3, page 46. Assume someone

spends $75 to buy 3 envelopes. The sample space describing the presence of $10 bills (H) 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:

_____________________________

Discrete Probability Distributions

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

P(X = 0) = ___________________

(recall results from last time)

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

x 0 1 2 3

P(X = x)

Another Example

• Example 3.3, pg 66

P(X = 0) =

_____________________

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)

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) = ________________

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)

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

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, __________

Homework due dates

• Friday, 9/3 pg.54-56• Wednesday, 9/8 pg. 72-74 (both sets)

(Check the web site on Thursday, 9/2, for an updated schedule.)