1 Chapter 8 Random Variables and Probability Distributions IRandom Sampling A.Population...

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Chapter 8

Random Variables and Probability Distributions

I Random Sampling

A. Population1. Population element

2. Sampling with and without replacement

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B. Random Sampling Procedures

1. Table of random numbers (from Appendix D.1)

1 2 3 4 5 6 7 8 9 . . . 21 22 23 24 25 1 10 27 53 96 23 71 50 54 36 . . . 26 78 25 47 47 2 28 41 50 61 88 64 85 27 20 94 76 62 11 89 3 34 21 42 57 02 4 61 81 77 23 23 5 61 15 18 13 54 6 91 76 21 64 64 7 00 97 79 08 06

59 19 18 97 82 82 11 54 16 86 20 26 44 91 13 32 37 30 28 59

84 97 50 87 46 42 34 43 39 28 52 01 63 01 59 56 08 25 70 29 30 19 99 85 48

50 87 41 60 76 83 44 88 96 07 . . . 30 56 10 48 59

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II Random 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.

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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.

HT 0 1

X 0 if coin is T1 if coin is H

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5. The random variable X is the name for any one ofa 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.

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B. Probability Distribution

1. Probability distribution for tossing a fair coin

X p(X = r)0 1/21 1/2

2. Graph of the probability distribution

.5

0 1

Prob

abili

ty

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3. Three-section T maze

G

S

4. Correct series of turns: R L R

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5. Number of ways of traversing the T maze:

2 2 2 = 8 (fundamental counting rule)_______________________

Turns Number of errors, X

R, L, R 0

R, R, R 1

R, L, L 1

L, L, R 1

R, R, L 2

L, R, R 2

L, L, L 2

L, R, L 3______________________________

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_________________________Possible Values of the

Random Variable X p(X = r)

0 .125

1 .375

2 .375

3 .125

Probability Distribution for Number of Errors in the Three-Choice T Maze

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6. Graph of the probability distribution for the Three-Choice T Maze

0 1 2 3

.1

.2.3.4

Number of errors

Prob

abili

ty

C. Expected Value of a Discrete Random Variable

E(X) = p(X1)X1 + p(X2)X2 + . . . + p(Xn)Xn =

where p(X1) + p(X2) + . . . + p(Xn) = 1

p X i i1

n X i

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1. For the T maze example, the expected value is

E(X) = p(X1)X1 + p(X2)X2 + p(X3)X3 + p(X4)X4

= .125(0) + .375(1) + .375(2) + .125(3) = 1.5

0 1 2 3

.1

.2.3.4

Number of errors

Prob

abili

ty

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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, Xi p(Xi) p(Xi)Xi

+ $35 1/38 1/38($35) = 35/38

– $1 37/38 37/38(–$1) = –37/38

n

iii XXpXE

1053.

382

3837

3835)()(

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D. Standard Deviation of a Discrete Random

Variable

p( X i ) X i E( X i )

2

1. For the T maze example, the standard deviation,

, is

.125(0 1.5)2 .375(1 1.5)2 .375(2 1.5)2 .125(3 1.5)2

= 0.866

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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.

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The probability that X will assume a value between a and

b is equal to the area under the curve between those two

points.

f ( X )

Xba

2. Distribution for a continuous random variable

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III Binomial 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

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2. Binomial function rule: probability of observing

exactly r heads (successes) in n trials is given by

p(X = r) = nCr prqn – r

where p(X = r) is the probability that the random

variable X equals r successes, nCr 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.

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3. Consider tossing n = five fair coins: the

probability of observing r = 0, 1, . . . , 5 heads is

p( X r) nCr

12

r12

n r

n!

r!(n r)!12

r12

n r

p( X 0) 5C0

12

012

5 0

5!

0!(5 0)!12

012

5 0

1

32

p( X 1) 5C1

12

112

5 1

5!

1!(5 1)!12

112

5 1

5

32

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p( X 2) 5C2

12

212

5 2

5!

2!(5 2)!12

212

5 2

1032

p( X 5) 5C5

12

512

5 5

5!

5!(5 5)!12

512

5 5

1

32

p( X 3) 5C3

12

312

5 3

5!

3!(5 3)!12

312

5 3

1032

p( X 4) 5C4

12

412

5 4

5!

4!(5 4)!12

412

5 4

5

32

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4. Binomial distribution for tossing five fair coins

0 1 32 4 5

2/324/426/328/32

10/32

Prob

abili

ty

Number of Heads, r

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C. Expected Value and Standard Deviation of a Binomial Random Variable

1. Expected value

E( X ) p( X i )X i

i1

n np

2. Standard deviation

p( X i ) X i E( X i )

2 npq

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

E( X ) np (5)(.5) 2.5

npq 5(1 / 2)(1 / 2) 1.118

0 1 32 4 5

2/324/426/328/32

10/32

Prob

abili

ty

Number of Heads, r

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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

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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

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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.