Defining Probabilities: Random Variables

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, ___________

2
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
• _____________________________

3
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)
4
Another Example

• Example 3.8, pg 80
• P(X 0)
• _____________________

5
Discrete Probability Distributions

• The discrete probability distribution function
  (pdf)
• f(x) P(X x) 0
• Sx f(x) 1
• The cumulative distribution, F(x)
• F(x) P(X x) St x f(t)

6
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)
  ________________

7
Another View

• The probability histogram

8
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
9
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, __________

10
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

11
Continuous Probability Distributions

• The continuous probability density function (pdf)
  
• f(x) 0, for all x ? R
• The cumulative distribution, F(x)

12
Probability Distributions

• Example Problem 3.7, pg. 88
• x, 0 lt x lt 1
• f(x) 2-x, 1 x lt 2
• 0, elsewhere
• 1st what does the function look like?
• P(X lt 120) ___________________
• P(50 lt X lt 100) ___________________

13
Your turn

• Problem 3.14, pg. 89