chapter 4: discrete random variables statistics. mcclave, statistics, 11th ed. chapter 4: discrete...

Chapter 4: Discrete Random Variables

Statistics

McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

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Where We’ve Been

Using probability to make inferences about populations

Measuring the reliability of the inferences


3

Where We’re Going

Develop the notion of a random variable

Numerical data and discrete random variables

Discrete random variables and their probabilities

4.1: Two Types of Random Variables

A random variable is a variable hat assumes numerical values associated with the random outcome of an experiment, where one (and only one) numerical value is assigned to each sample point.

4McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables


A discrete random variable can assume a countable number of values. Number of steps to the top of the Eiffel Tower*

A continuous random variable can assume any value along a given interval of a number line. The time a tourist stays at the top

once s/he gets there

*Believe it or not, the answer ranges from 1,652 to 1,789. See Great Buildings



Discrete random variables Number of sales Number of calls Shares of stock People in line Mistakes per page

Continuous random variables Length Depth Volume Time Weight


4.2: Probability Distributions for Discrete Random Variables

The probability distribution of a discrete random variable is a graph, table or formula that specifies the probability associated with each possible outcome the random variable can assume. p(x) ≥ 0 for all values of x p(x) = 1


4.2: Probability Distributions for Discrete Random Variables

Say a random variable x follows this pattern: p(x) = (.3)(.7)x-1

for x > 0. This table gives the

probabilities (rounded to two digits) for x between 1 and 10.

x P(x)

1 .30

2 .21

3 .15

4 .11

5 .07

6 .05

7 .04

8 .02

9 .02

10 .01


4.3: Expected Values of Discrete Random Variables

The mean, or expected value, of a discrete random variable is

( ) ( ).E x xp x



The variance of a discrete random variable x is

The standard deviation of a discrete random variable x is

2 2 2[( ) ] ( ) ( ).E x x p x

2 2 2[( ) ] ( ) ( ).E x x p x


)33(

)22(

)(

xP

xP

xP

Chebyshev’s Rule Empirical Rule

≥ 0 .68

≥ .75 .95

≥ .89 1.00





In a roulette wheel in a U.S. casino, a $1 bet on “even” wins $1 if the ball falls on an even number (same for “odd,” or “red,” or “black”).

The odds of winning this bet are 47.37%

9986.

0526.5263.1$4737.1$

5263.)1$(

4737.)1$(

loseP

winP

On average, bettors lose about a nickel for each dollar they put down on a bet like this.(These are the best bets for patrons.)

4.4: The Binomial Distribution

A Binomial Random Variable n identical trials Two outcomes: Success or Failure P(S) = p; P(F) = q = 1 – p Trials are independent x is the number of Successes in n trials



A Binomial Random Variable n identical trials Two outcomes: Success

or Failure P(S) = p; P(F) = q = 1 – p Trials are independent x is the number of S’s in n

trials

Flip a coin 3 times

Outcomes are Heads or Tails

P(H) = .5; P(F) = 1-.5 = .5

A head on flip i doesn’t change P(H) of flip i + 1



Results of 3 flips Probability Combined Summary

HHH (p)(p)(p) p3 (1)p3q0

HHT (p)(p)(q) p2q

HTH (p)(q)(p) p2q (3)p2q1

THH (q)(p)(p) p2q

HTT (p)(q)(q) pq2

THT (q)(p)(q) pq2 (3)p1q2

TTH (q)(q)(p) pq2

TTT (q)(q)(q) q3 (1)p0q3



The Binomial Probability Distribution p = P(S) on a single trial q = 1 – p n = number of trials x = number of successes

xnxqpx

nxP

)(



The Binomial Probability Distribution

xnxqpx

nxP

)(


Say 40% of the class is female.

What is the probability that 6 of the first 10 students walking in will be female?


1115.

)1296)(.004096(.210

)6)(.4(.6

10

)(

6106

xnxqpx

nxP



Mean

Variance

Standard Deviation

A Binomial Random Variable has


19

2

np

npq

npq


16250

2505.5.1000

5005.10002

npq

npq

np

For 1,000 coin flips,

The actual probability of getting exactly 500 heads out of 1000 flips is just over 2.5%, but the probability of getting between 484 and 516 heads

(that is, within one standard deviation of the mean) is about 68%.


4.5: The Poisson Distribution

Evaluates the probability of a (usually small) number of occurrences out of many opportunities in a … Period of time Area Volume Weight Distance Other units of measurement



!)(

x

exP

x

= mean number of occurrences in the given unit of time, area, volume, etc.

e = 2.71828…. µ = 2 =



1008.!5

3

!)5(

35

e

x

exP

x

Say in a given stream there are an average of 3 striped trout per 100 yards. What is the probability of seeing 5 striped trout in the next 100 yards, assuming a Poisson distribution?



0141.!5

5.1

!)5(

5.15

e

x

exP

x

How about in the next 50 yards, assuming a Poisson distribution? Since the distance is only half as long, is only

half as large.


4.6: The Hypergeometric Distribution

In the binomial situation, each trial was independent. Drawing cards from a deck and replacing

the drawn card each time If the card is not replaced, each trial

depends on the previous trial(s). The hypergeometric distribution can be

used in this case.


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26

Randomly draw n elements from a set of N elements, without replacement. Assume there are r successes and N-r failures in the N elements.

The hypergeometric random variable is the number of successes, x, drawn from the r available in the n selections.



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n

N

xn

rN

x

r

xP )(

where N = the total number of elementsr = number of successes in the N elementsn = number of elements drawnX = the number of successes in the n elements



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n

N

xn

rN

x

r

xP )(

)1(

)()(2

2

NN

nNnrNrN

nr


44.22.2)2()2()2or2(

22.45

)1)(10(

2

10

22

510

2

5

)2()2(

FPMPFMP

FPMP

Suppose a customer at a pet store wants to buy two hamsters for his daughter, but he wants two males or two females (i.e., he wants only two hamsters in a few months)

If there are ten hamsters, five male and five female, what is the probability of drawing two of the same sex? (With hamsters, it’s virtually a random selection.)


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chapter 4: discrete random variables statistics. mcclave, statistics, 11th ed. chapter 4: discrete...

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