1. EGR 252 - 41 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. EGR 252 - 42 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. EGR 252 - 43 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: x0123 P(X = x)

4. EGR 252 - 44 Another Example Example 3.8, pg 80 P(X = 0) = _____________________

5. EGR 252 - 45 Discrete Probability Distributions The discrete probability distribution function (pdf) –f(x) = P(X = x) ≥ 0 –Σ x f(x) = 1 The cumulative distribution, F(x) –F(x) = P(X ≤ x) = Σ t ≤ x f(t)

6. EGR 252 - 46 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. EGR 252 - 47 Another View The probability histogram

8. EGR 252 - 48 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. xP(x)P(x) 012012

9. EGR 252 - 49 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. EGR 252 - 410 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. EGR 252 - 411 Continuous Probability Distributions The continuous probability density function (pdf) f(x) ≥ 0, for all x ∈ R The cumulative distribution, F(x)

12. EGR 252 - 412 Probability Distributions Example: Problem 3.7, pg. 88 x, 0 < x < 1 f(x) =2-x,1 ≤ x < 2 0, elsewhere 1 st – what does the function look like? a)P(X < 120) = ___________________ b)P(50 < X < 100) = ___________________ {

13. EGR 252 - 413 Your turn Problem 3.14, pg. 89