1. . 6- 1 © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. Chapter Six McGraw-Hill/Irwin Discrete Probability Distributions

2. Random variable A result from an experiment that, by chance, can take different values. Egs. 1. No. of heads you would get in 3 tosses of a coin 2. No. of employees absent on a shift 3. No. of students at CSUN in a semester 4. No. of minutes to drive home from CSUN 5. Inches of rainfall in LA during a year 6. Tire pressure in PSI of a car tire Examples 1-3 are ‘discrete’ 4-6 are ‘continuous’ random variables In this chapter, we focus on the ‘discrete’.

3. Probability Distribution A listing of all possible outcomes of an experiment and the corresponding probability.

4. = 1/8 = 3/8 = 1/8 c P F T N Possible Outcomes

5. Note the similarity to histogram Probability Distribution

6. Mean of Discrete Probability Distribution Mean of Discrete Probability Distribution where  represents the mean ox is each outcome oP(x) is the probability of the various outcomes x.

7. μ = 0 *.125 + 1 *.375 + 2 *.375 + 3 *.125 = 1.5 (which by intuition makes sense because for each toss you have 50% chance of getting a head). Calculating Mean/Expected Value of Heads in 3 coin-toss experiment  is a weighted average. It is also referred to as Expected Value, E(X).

8. Binomial Probability Distribution An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure (bi means two). The data collected are the results of counts (hence, a discrete probability distribution). The probability of success stays the same for each trial (independence).

9. n! x!(n-x)! n is the number of trials x is the number of observed successes π is the probability of success on each trial Binomial Probability Distribution Can you logically explain this formula?

10. Let us re-visit the 3-toss coin experiment n = 3 π =.5 X = 0,1,2,3 (number of heads) P(0) = 3 C 0 * (.5) 0 * (1 -.5) 3-0 =.125 P(1) = 3 C 1 * (.5) 1 * (1 -.5) 3-1 =.375 P(2) = 3 C 2 * (.5) 2 * (1 -.5) 3-2 =.375 P(3) = 3 C 3 * (.5) 3 * (1 -.5) 3-3 =.125 You can also use Appendix A (page 489)

11. Practice time! Do Self-Review 6-3, Page 168 Use Appendix A, Page 490

12. Cumulative Binomial Probability Distribution Problem 21, Page 173 Use table in Page 491 a. 0.387 ( n=9; x=9; straight from table ) b. 0.001 ( P(x<5) = P(x ≤ 4) ) c. 0.992 ( 1 – P(x ≤ 5) = 1 – 0.008 ) d. 0.946 [ (P(x=7) + (P(x=8) + (P(x=9) ]

13. Mean of the Binomial Distribution Logic: If you throw a coin say 100 times, how many times you would expect to get a head? For each throw, the probability of getting a head is.5. So, in 100 trials, you would expect 50 heads (which is 100*.5; ie. nπ )

14. Variance of the Binomial Distribution For the previous example, = 100*.5*(1-.5) σ 2 = 25 S.D. σ = 5