Chapter 5 Discrete Probability Distribution

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Chapter 5 Discrete Probability Distribution

• I. Basic Definitions
• II. Summary Measures for Discrete Random Variable
• Expected Value (Mean)
• Variance and Standard Deviation
• III. Two Popular Discrete Probability
  Distributions
• Binomial Distribution
• Poisson Distribution

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• I. Basic Definitions
• Random Variable (p.195)
• a numerical description of the outcomes of an
  experiment.
• Assign values to outcomes of an experiment so
  the experiment can be represented as a random
  variable.
• Example
• Test scores 0 ? X ? 100
• Toss coin X 0, 1
• Roll a die X 1, 2, 3, 4, 5, 6

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• I. Basic Definitions
• Discrete Random Variable It takes a set of
  discrete values. Between two possible values
  some values are impossible.
• Continuous Random Variable Between any two
  possible values for this variable, another value
  always exists.
• Example
• Roll a die X 1, 2, 3, 4, 5, 6. X is a
  discrete random variable.
• Weight of a person selected at random X is a
  continuous random variable.

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• I. Basic Definitions
• Two ways to present a discrete random variable
• Probability Distribution (p.198 Table 5.3)
• a list of all possible values (x) for a random
  variable and probabilities (f(x)) associated
  with individual values.
• Probability Function probability may be
  represented as a function of values of the
  random variable.
• Example p.199
• Consider the experiment of rolling a die and
  define the random variable X to be the number
  coming up.
• 1. Probability distribution
• 2. Probability function f(x) 1/6.

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• I. Basic Definition
• Valid Discrete Probability Function
• 0 ? f(x) ? 1 for all f(x) AND
• ? f(x) 1
• Example p.200 7
• a. The probability distribution is proper because
  all f(x) meet requirements 0 ? f(x) ? 1 and ?
  f(x) 1.
• b. P(X30) f(30) .25
• c. P(X?25) f(25) f(20) .15 .20 .35
• d. P(Xgt30) f(35) .40
• Homework p.201 10, p.201 14

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I. Basic Definitions Example p.202 14
a. Find valid f(200) .1.2.3.25.1f(200)
1, so f(200) .05 b. P(?) P(Xgt0)
f(50)f(100)f(150)f(200) .7 c.
P(?) P(X?100) f(100)f(150)f(200) .4 (at
least)
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• II. Summary Measures for Discrete Random Variable
• Expected Value (mean) E(X) or ? (p.203)
• E(X) ?xf(x)
• Variance Var(X) or ?2 (p.203)
• Var(X) ?(x- ? )2f(x)
• Homework p.204 16
• Example Consider the experiment of tossing coin
  and define X to be 0 if head and 1 if
  tail. Find the expected value and
  variance.

E(X) ?xf(x) (0)(.5)(1)(.5) .5 Var(X)
?(x- ? )2f(x) (0 - .5)2(.5)(1 - .5)2(.5)
.25
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III. Two Popular Discrete Probability
Distributions Binomial and
Poisson Outlines 1. Probability Distribution
Binomial Distribution Table 5 (p.989 - p.997).
Given n, p, x ? f(x). Poisson Distribution
Table 7 (p.999 - p.1004). Given ?, x ?
f(x). 2. Applications Difference between
Binomial and Poisson. 3. Applications
criterion to define success.
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• III. Two Popular Discrete Probability
  Distributions
• 1. Binomial Distribution p.207
• Random variable X x successes out of n trials
  (the number of successes in n trials).
• Three conditions for Binomial distribution
• n independent trials
• Two outcomes for each trial success and
  failure.
• p probability of a success is a constant from
  trial to trial.

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III. Two Popular Discrete Probability
Distributions Example Toss a coin 10 times. We
define the success as a head and X is the
number of heads from 10 trials. Does X
have a binomial distribution? Answer
Yes. Follow-up Why? n? p? Example Roll a die 10
times. We define the success as 5 or
more points coming up and X is the
number of successes from 10 trials. Does X
have a binomial distribution? Answer Yes.
Why? n? p?
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• III. Two Popular Discrete Probability
  Distributions
• Summary Measures for Binomial Distribution
• E(X) np p.214
• Var(X) np(1-p) p.214
• Binomial Probability Distribution p.212
• f(x) P(Xx)
• Table 5 (p.989-p.997) Given n, p and
  x, find f(x).
• Homework p.216 26, 27, 29, 30 c, d.

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Example p.216 25 Given
Binomial, n 2, p .4 Success X of
successes in 2 trials Answer b. f(1) ? (Table
5) f(1) .48 c. f(0) ? f(0) .36 d.
f(2) ? e. P(X?1) ? f(2) .16)
P(X?1) f(1) f(2) .64 e. E(X) np .8
Var(X) np(1-p) .48 ? ?
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Example p.217 35 (Application) Binomial
distribution? 1. Is there a criterion for
success and failure? 2. n trials? n? 3.
p? Given
Success withdraw n 20 p .20 Answer a.
P(X ? 2) f(0) f(1) f(2) .0115
.0576 .1369 .2060 b. P(X4) f(4)
.2182 c. P(Xgt3) 1 - P(X ? 3) 1 f(0)
f(1) f(2) f(3) 1 - .0115
- .0576 - .1369 - .2054 .5886 d. E(X) np
(20)(.2) 4.
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• III. Two Popular Discrete Probability
  Distributions
• 2. Poisson Distribution p.218
• Random variable X x occurrences per unit (the
  number of occurrences in a unit - time, size,
  ...).
• Example X the number of calls per hour. Does X
  have a Poisson distribution?
• Answer Yes. Because
• Occurrence a call.
• Unit an hour.
• X the number of occurrences (calls) per unit
  (hour).
• Reading p.216 29, 30, and p.220 40, p.221 42
• Binomial or Poisson? If Binomial, success? p?
  n?
• If Poisson, occurrence? ?? unit?

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• III. Two Popular Discrete Probability
  Distributions
• Summary Measures for Poisson Distribution
• E(X) ? (? is given)
• Var(X) ?
• Poisson Probability Distribution p.219
• f(x) P(Xx)
• Table 7 (p.999 through p.1004) Given ?
  and x, find f(x).
• Note Keep the unit of ? consistent with the
  question.
• Homework p.220 38, p.221 42, 43

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• Example p.220 39
• Given
• Poisson distribution, because
• X the number of occurrences per time period.
• ? 2 and unit is a time period.
• Note Unit in questions may be different.
• Answer
• a. f(x)
• b. What is the average number of occurrences in
• three time periods? (Different unit!)
• ?3 (3)(? ) 6. c. f(x)
• d. f(2) P(X2) .2707 (Table 7. ? 2, x 2)
• e. f(6) P(X6) .1606 (Table 7. ? 6, x 6)

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Example p.221 43 Given

• Poisson distribution, because
• X the number of arrivals per time period.
• ? 10 and unit is per minute.
• Note Unit in questions may be different.
• Answer
• a. f(0) 0 (Table 7. ? 10, x 0)
• b. P(X?3) f(0)f(1)f(2)f(3) (Table 7. ?
  10, x ?)
• 0 .0005 .0023 .0076 .0104
• c. P(X0) f(0) .0821
• (Table 7. ? (10)(15/60)2.5 for unit of 15
  seconds, x 0)
• d. P(X?1) ? (Table 7. ? 2.5, x ?)
• P(X?1) f(1) f(2) f(3) ???
• 1 - P(Xlt1) 1 - f(0) 1 - .0821
  .9179.

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• Chapter 5 Summary
• Binomial distribution X of successes out
  of n trials
• Poisson distribution X of occurrences per
  unit
• For Binomial distribution, E(X)np,
  Var(X)np(1-p)
• For Poisson distribution, E(X) ? Var(X)
• Binomial distribution table Table 5
• Poisson distribution table Table 7
• Poisson Approximation of Binomial distribution
• If p ? .05 AND n ? 20, Poisson distribution
  (Table 7) can
• be used to find probability for Binomial
  distribution.
• Example A Binomial distribution with n 250 and
  p .01,
• f(3) ?
• Answer ? np (250)(.01)2.5. From Table 7,
  f(x).2138
• Sampling with replacement and without
  replacement.

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Example A bag of 100 marbles contains 10 red
ones. Assume that samples are drawn randomly
with replacement. a. If a sample of two is
drawn, what is the probability that both
will be red? b. If a sample of three is drawn,
what is the probability that at least one will
be red? (without replacement Hypergeometric
distribution p.214) Answer
a. n 2, p .1, f(2) ? From Table 5, f(2)
.01 b. n 3, p .1, P(X?1) f(1) f(2)
f(3) From Table 5, P(X?1) .2430 .0270
.0010 .271