Chapter 5 Discrete and Continuous Probability Distributions

1
Chapter 5 Discrete and Continuous Probability
Distributions

• GOALS Upon successful completion, you should be
  able to
• Identify binomial random variables
• Use the binomial distribution to determine
  probabilities
• Identify hypergeometric random variable
• Use the hypergeometric distribution to calculate
  probabilities
• Use the normal distribution to calculate
  probabilities

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The Binomial Probability Distribution

• Properties of a Binomial Experiment
• Sequence of n identical trials.
• Two outcomes, success and failure, are possible
  on each trial.
• The probability of a success, denoted by p, does
  not change from trial to trial.
• The trials are independent.

3
Example Company Insurance

• Company records indicate that 90 of the
  employees annually use the companys health
  insurance plan.
• Suppose 3 employees are randomly selected and
• x the number using the insurance.
• Is x a binomial random variable?

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Example Company Insurance

• Binomial Probability Properties Apply?
• 3 identical trials n3
• Two outcomes, success use insurance Y and
  failure do not use insurance N, are possible
  on each trial.
• The probability of a success, denoted by p .90,
  does not change from trial to trial.
• The trials are independent?

5
Example Company Insurance

• So, what is the probability that just 1 of the 3
  employees will use the company insurance plan?
• That is,
• P(x1 n3, p.90) ?

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Example Company Insurance Tree Diagram
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Example Company Insurance

• P( N1N2Y3 N1Y2N3 Y1N2N3 )
• (.1)(.1)(.9)
• (.1)(.9)(.1)
• (.9)(.1)(.1) .027

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The Binomial Probability Function

• where
• P(x) the probability of x successes in n
  trials
• n the number of trials
• p the probability of success on any one
  trial
• q 1 - p

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Example Company Insurance

• Using the Binomial Probability Function
• (3)(0.90)(0.01)
• .027

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Example Company Insurance

• Using the Tables of Binomial Probabilities
• see page 731

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The Binomial Probability Distribution

• Expected Value E(x) ? n p
• Variance Var(x) ? 2 n p q
• Standard Deviation

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Example Company Insurance

• Expected Value
• E(x) ? 3(.9) n p 2.7 employees
• Variance
• ? 2 n p q 3(.9)(.1) .27
• Standard Deviation

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The Hypergeometric Probability Distribution

• Closely related to the binomial distribution.
• Trials are not independent, and the probability
  of success changes from trial to trial.

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The Hypergeometric Probability Function

• for 0
• where
• f(x) probability of x successes in n
  trials
• n number of trials
• N number of elements in the population
• r number of elements in the
  population labeled success

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Florida Lotto Ticket - Back
16
Florida Lotto 5 of 6
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Normal Distribution Characteristics

• Symmetric and Bell-shaped
• Location determined by the mean, m, which can be
  any numerical value -, 0,
• Spread determined by the standard deviation, s
• Mean, median and mode at the center
• Area under curve equivalent to probability
• Total area 1 .5 on each side

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Normal Probability Distribution

• where
• ? mean
• ? standard deviation
• ? 3.14159
• e 2.71828

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Normal Curves Different ?, Same ?
s 0.5
s 0.5
? 12
? 20
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Normal Curves Same ?, Different ?
s 0.5
s 1.0
? 20
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Recall the Empirical Rule

• µ 1s ? About 68 of the data
• µ 2s ? About 95 of the data
• µ 3s ? About 99.7 of the data

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Empirical Rule

• So, if µ 10 and s 2,
• then 68 of the data will be between 8 and 12
• then 95 of the data will be between 6 and 14
• then 99.7 of the data will be between 4 and 16

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What about other probabilities?

• If µ 10 and s 2,what percentage of the data
  will be between 7 and 13?

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What about other probabilities?

• If µ 10 and s 2, what percentage of the data
  will be between 7 and 13?
• Solution, integrate
• from 7 to 13.

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Calculating Z-scores
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Standard Normal Distribution

• Normal Distribution with
• µ 0 and s 1
• Also called the z-distribution.

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Standard Normal Distribution

• Integrate the z-distribution ? table of
  probabilities
• Convert other normal distributions to the
  z-distribution to determine probabilities.

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Standard Normal Table

• Table of probabilities from the mean to some
  positive z-value page 743

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Calculating Normal Probability

• GE fluorescent light bulbs have an average life
  of 1000 hours with a standard deviation of 50
  hours.
• What percentage of light bulbs last between 1000
  hours and 1080 hours?

1000 1080
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Calculating Normal Probability
1000 1080
X
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Calculating Normal Probability
What percentage of light bulbs last more than
1080 hours?
.5000 - .4452 .0548
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Calculating Normal Probability
What percentage of light bulbs last less than
1080 hours?
.5000 .4452 .9452
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Calculating Normal Probability
What percentage of light bulbs last between than
1080 and 1100 hours?
.4772 - .4452 .0320
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Finding a Percentile

• DieHard makes a car battery with an average life
  of 96 months and a standard deviation of 6
  months.
• What is the value that separates the 5 of the
  batteries that have the shortest lifespan?
• What is the 5th percentile?

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Finding a Percentile

• Here we know µ and s. We are looking for the
  value of x that corresponds to the 5th
  percentile.

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Finding a Percentile

• Find the 80th percentile.

X
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• End
• of
• Chapter 5