Chapter 4 Probability Distributions

1
Chapter 4Probability Distributions

• 4-1 Random Variables
• 4-2 Binomial Probability Distributions
• 4-3 Mean, Variance, Standard Deviation for
  the Binomial Distribution
• 4-4 Other Discrete Probability Distributions

2
Overview

• This chapter will deal with the construction of
• probability distributions
• by combining the methods of Chapter 2 with the
  those of Chapter 3.
• Probability Distributions will describe what will
  probably happen instead of what actually did
  happen.

3
Combining Descriptive Statistics Methods and
Probabilities to Form a Theoretical Model of
Behavior
4
4-1

• Random Variables

5
Definitions

• Random Variable
• a variable (typically represented by x) that has
  a single numerical value, determined by chance,
  for each outcome of a procedure
• Probability Distribution
• a graph, table, or formula that gives the
  probability for each value of the random variable

6
Probability DistributionNumber of Girls Among
Fourteen Newborn Babies
x
P(x)
0.000 0.001 0.006 0.022 0.061 0.122 0.183 0.209 0.
183 0.122 0.061 0.022 0.006 0.001 0.000

• 0
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12
• 13
• 14

• Questions
• Let X of girls
• P(X3) ?
• P(X 2) ?
• P(X 1) ?

7
Probability Histogram
8
Definitions

• Discrete random variable
• has either a finite number of values or
  countable number of values, where countable
  refers to the fact that there might be infinitely
  many values, but they result from a counting
  process.
• Continuous random variable
• has infinitely many values, and those values can
  be associated with measurements on a continuous
  scale with no gaps or interruptions.

9
Requirements for Probability Distribution
?

• P(x) 1
• where x assumes all possible values
• 0 ? P(x) ? 1
• for every value of x

See 3 on hw
10
Mean, Variance and Standard Deviation of a
Probability Distribution

• Formula (Mean)
• µ ? x P(x)
• Formula (Variance)
• ?2 ? (x - µ)2 P(x)
• Formula (Standard Deviation)
• ? (x - µ)2 P(x)

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Roundoff Rule for µ, ?2, and ?

• Round results by carrying one more decimal
  place than the number of decimal places used for
  the random variable x. If the values of x are
  integers, round µ, ??2, and ?? to one decimal
  place.

12
TI-83 Calculator

• Calculate Mean and Std. Dev from a Probability
  Distribution
• Press Stat
• Press 1 Edit
• Enter values of random variable (x) in L1
• Enter probability P(x) in L2
• Press Stat
• Cursor over to CALC
• Choose the 1-Var stats option
• Enter 1-Var stats L1,L2

13
Using Excel

• See Probability Distribution Worksheet
• Examples
• See Introduction to probability distributions
  handout
• Go to Excel (dice example class assignment)
• Find missing probability, mean, SD and unusual
  values (test question see 6 on hw)

14
Unusual and Unlikely Values

• Unusual if greater than 2 standard deviations
  from the mean, that is x 2? and x 2s
• Unlikely if probability is very small, usually
  less than .05. This is consistent with the 2?
  idea associated with the empirical rule.
• 6f and 7c on HW

15
Definition

• Expected Value
• The average value of outcomes
• E ? x P(x)

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E ? x P(x)
Example 9 on hw
Event Win Lose
x 5 - 5
P(x) 244/495 251/495
x P(x) 2.465 - 2.535
E -.07
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4-2

• Binomial Random Variables

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Binomial Random Variables

• Facts
• Discrete (we can count the outcomes)
• Have to do with random variables having 2
  outcomes. Examples heads/tails, boy/girl,
  yes/no, defective/not defective, etc.
• A binomial distribution is the sum of several
  trials. Example of heads when a coin is
  tossed three times

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Notation for Binomial Probability Distributions

• n fixed number of trials
• x specific number of successes in n trials
• p probability of success in one of n trials
• q probability of failure in one of n trials
• (q 1 - p )
• P(x) probability of getting exactly x successes
  among n trials

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Binomial Probability Formula
Method 1
n !

• P(x) px qn-x

(n - x )! x!

• P(x) nCx px qn-x

for calculators use nCr key, where r x
21
Binomial Probability Formula
n !
P(x) px qn-x
(n - x )! x!
Probability of x successes among n trials for any
one particular order
Number of outcomes with exactly x successes
among n trials
22

• Example Toss a coin 3 times.
• Let x number of heads and find
• P(2)
• P(at least 2)

• This is a binomial experiment so you need to know
  4 things p, q, n and x.
• p.5
• q.5
• n3
• a) x 2 b) x 0 then 1 then 2

On the test you will have to construct the entire
probability distribution for tossing a coin n
times and observing the number of heads. Should
do this before the test.
23
Example Find the probability of getting exactly
2 correct responses among 5 different requests
from directory assistance. Assume in general,
they are correct 80 of the time.

• This is a binomial experiment where
• n 5
• x 2
• p 0.80
• q 0.20
• Using the binomial probability formula to solve
• P(2) 5C2 0.8 0.2 0.0512

3
2
24
Binomial Table
Method 2
Two tables are available on website
25
Example Using Table for n 5 and p 0.80,
find the following a) The probability
of exactly 2 successesb) The probability of at
most 2 successesc) The probability of at least
1 success
Test Question

• a) P(2) 0.0512
• b) P(at most 2) P(0) or P(1) or P(2)
• 0.0003 0.0064 0.0512
• 0.0579
• c) P(at least 1) 1 P(0) 1 .0003 .9997

26
Using Technology
Method 3

• Calculator function (TI-83)
• See binomial distribution worksheet
• See also coin example worksheet

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TI-83 Calculator

• Finding Binomial Probabilities (complete
  distribution)
• Press 2nd Distr
• Choose binopdf
• Enter binopdf(n,p)
• Press STO L2 (stores probabilities in column L2)
• Press Stat
• Choose Edit (to view probabilities)
• Optional enter the values of the random
  variable in L1

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TI-83 Calculator

• Finding Binomial Probabilities (individual value)
• Press 2nd Distr
• Choose binopdf
• Enter binopdf(n,p,x)

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TI-83 Calculator

• Finding Binomial Probabilities (cummulative)
• Press 2nd Distr
• Choose binocdf
• Enter binocdf(n,p,x)
• This yields the sum of the probabilities
  from 0 to x.
• Example
• Let n6 and p0.2
• P(Xlt3) binocdf(6,.2,3)

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4.3 Mean, variance and standard deviation of a
Binomial Probability Distribution
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For Any Discrete Probability Distribution the
general formulas are

• µ ?x P(x)
• ??2???? ? (x - µ) 2 P(x)

• ?? ? (x
  - µ) 2 P(x)

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For Binomial Distributions

• µ n p
• ??2? n p q

• ???? n p q

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Example Find the mean and standard deviation
for students that guess answers on a multiple
choice test with 5 answers and 20 questions.

• We previously discovered that this scenario could
  be considered a binomial experiment where
• n 20
• p 0.2
• q 0.8
• Using the binomial distribution formulas
• µ (20)(0.2) 4 correct answers
• ?? (20)(0.2)(0.8) 1.8 answers
  (rounded)

Test question
34
Reminder

• Maximum usual values µ 2 ?
• Minimum usual values µ - 2 ?

35
Example Determine whether guessing 7 correct
answers is unusual.

• For this binomial distribution,
• µ 4 answers
• ?? 1.8 answers
• µ 2 ? 4 2(1.8) 7.6
• µ - 2 ? 4 - 2(1.8) .4
• The usual number of correct answers would be
  from .4 to 7.6, so guessing 7 correct answers
  would not be an unusual result.

Test question
36
4.4 Other Discrete Probability Distributions

• Poisson
• Geometric
• Hypergeometric
• Negative Binomial
• And more

37
Poisson Distribution

• Definition
• a discrete probability distribution that
  applies to occurrences of some event over a
  specific interval.

Will be a question on the test for you to
differentiate between a binomial and a poisson
distribution
38
Definition

• Poisson Distribution
• a discrete probability distribution that
  applies to occurrences of some event over a
  specific interval.
• Probability Formula

39
Example Look at 1

• Why is this a poisson distribution?
• µ 5
• We need to find various probabilities using
• Lets find P(7)
• Look at the Poisson function in Excel

µ x e -µ
P(x)
x!
40
Geometric Distribution

• Definition
• a discrete probability distribution of the
  number of trials needed to get one success.

Will be a question on the test for you to
differentiate between a binomial, poisson and a
geometric distribution
41
Geometric Distribution

• Example
• Roll a die 5 times. What is the probability
  of getting your first 2 on the 5th roll.

42
Negative Binomial Distribution

• Definition
• a discrete probability distribution of the
  number of trials needed to get a get a specified
  number of successes.

43
Negative Binomial Distribution

• Example
• a basketball player has a 70 chance of making
  a free throw, what is the probability of making
  his 3rd free throw on his 5th shot.

44
Hypergeometric Distribution

• Hypergeometric Experiment
• A sample of size n is randomly selected without
  replacement from a population of N items.
• . In the population, k items can be classified as
  successes, and N - k items can be classified as
  failures.

45
Hypergeometric Distribution

• Notation
• N The number of items in the population.
• k The number of items in the population that are
  classified as successes.
• n The number of items in the sample.
• x The number of items in the sample that are
  classified as successes.
• kCx The number of combinations of k things,
  taken x at a time.

46
Hypergeometric Distribution

• Example
• Suppose we randomly select 5 cards without
  replacement from an ordinary deck of playing
  cards. What is the probability of getting exactly
  2 red cards (i.e., hearts or diamonds)?
• P kCx N-kCn-x / NCn 26C2
  26C3 / 52C5 325 2600 /
  2,598,960 0.32513