1. 1 Chapter 8 Random Variables and Probability Distributions IRandom Sampling A.Population 1.Population element 2.Sampling with and without replacement

2. 2 B.Random Sampling Procedures 1.Table of random numbers (from Appendix D.1)

3. 3 IIRandom Variables and Their Distributions A.Random Variable: A Numerically Valued Function Defined on a Sample Space 1.A function consists of two sets of elements and a rule that assigns to each element in the first set one and only one element in the second set. 2.Examples: {(a, 1), (b, 5), (c, 6)} {(Mike, tall), (Jim, short), (Joe, medium)} 3.If the second element is a number, the function is numerically valued.

4. 4 4.A random variable associates one and only one number with each point in a sample space; thus, it is a numerically valued function defined on a sample space. Example: consider tossing a fair coin; points in the sample space can be associated with numbers on the real number line.

5. 5 5.The random variable X is the name for any one of a set of permissible numerical values of a random experiment. 6.Discrete random variable: range can assume only a finite number of values or an infinite number of values that is countable. 7.Continuous random variable: range is uncountably infinite.

6. 6 B.Probability Distribution 1.Probability distribution for tossing a fair coin Xp(X = r) 01/2 11/2 2.Graph of the probability distribution

7. 7 3.Three-section T maze 4. Correct series of turns: R L R

8. 8 5.Number of ways of traversing the T maze: 2  2  2 = 8 (fundamental counting rule) _______________________ TurnsNumber of errors, X R, L, R0 R, R, R1 R, L, L1 L, L, R1 R, R, L2 L, R, R2 L, L, L2 L, R, L3 ______________________________

9. 9 _________________________ Possible Values of the Random Variable Xp(X = r) 0.125 1.375 2.375 3.125 Probability Distribution for Number of Errors in the Three-Choice T Maze

10. 10 6.Graph of the probability distribution for the Three- Choice T Maze C.Expected Value of a Discrete Random Variable E(X) = p(X 1 )X 1 + p(X 2 )X 2 +... + p(X n )X n = where p(X 1 ) + p(X 2 ) +... + p(X n ) = 1

11. 11 1.For the T maze example, the expected value is E(X) = p(X 1 )X 1 + p(X 2 )X 2 + p(X 3 )X 3 + p(X 4 )X 4 =.125(0) +.375(1) +.375(2) +.125(3) = 1.5

12. 12 2.Expected value of a bet at the roulette table; you pay $1.00 to win $35.00. The wheel has 38 slots. Possible Winnings, X i p(X i )p(X i )X i + $351/381/38($35) = 35/38 – $137/3837/38(–$1) = –37/38

13. 13 D.Standard Deviation of a Discrete Random Variable 1.For the T maze example, the standard deviation, , is = 0.866

14. 14 E.Expected Value of a Continuous Random Variable 1.A continuous random variable can assume an infinite number of values. The probability that a continuous random variable, X, has a particular value is zero. Hence, we refer to the probability that X lies in an interval between two values of the random variable.

15. 15 The probability that X will assume a value between a and b is equal to the area under the curve between those two points. 2.Distribution for a continuous random variable

16. 16 IIIBinomial Distribution A. Three Characteristics of a Bernoulli Trial B.Binomial Distribution 1.Binomial random variable: number of successes observed on n ≥ 2 identical Bernoulli trials

17. 17 2.Binomial function rule: probability of observing exactly r heads (successes) in n trials is given by p(X = r) = n C r p r q n – r where p(X = r) is the probability that the random variable X equals r successes, n C r is the combination of n objects taken r at a time, p is the probability of a success, and q = 1 – p is the probability of a failure.

18. 18 3.Consider tossing n = five fair coins: the probability of observing r = 0, 1,..., 5 heads is

19. 19

20. 20 4.Binomial distribution for tossing five fair coins

21. 21 C.Expected Value and Standard Deviation of a Binomial Random Variable 1. Expected value 2.Standard deviation

22. 22 3.For the coin tossing experiment where n = 5 and p = q = 1/2

23. 23 D.Binomial Model Is Appropriate Under the Following Conditions 1.There are n trials involving a population whose elements belong to one of two classes 2.Probability of obtaining an element remains constant from trial to trial, as when sampling with replacement from a finite population 3.Outcomes of successive trials are independent

24. 24 E.Two Other Models 1.Multinomial distribution is an extension of the binomial distribution for the case in which there are more than two classes. It is identical to the binomial distribution when there are only two classes. Probability of obtaining an element remains constant from trial to trial, as when sampling with replacement Outcomes of successive trials are independent

25. 25 2.Hypergeometric distribution is appropriate for the case in which there are more than two classes but the probabilities associated with the classes do not remain constant as when sampling without replacement from a finite population.