Special Discrete Distributions

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Special Discrete Distributions
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Binomial Distribution B(n,p)

• Each trial results in one of two mutually
  exclusive outcomes. (success/failure)
• There are a fixed number of trials
• Outcomes of different trials are independent
• The probability that a trial results in success
  is the same for all trials
• The binomial random variable x is defined as the
  number of successes out of the fixed number

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Are these binomial distributions?

• Toss a coin 10 times and count the number of
  heads
• Yes
• Deal 10 cards from a shuffled deck and count the
  number of red cards
• No, probability does not remain constant
• Two parents with genes for O and A blood types
  and count the number of children with blood type
  O
• No, no fixed number

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Toss a 3 coins and count the number of
heads Find the discrete probability distribution
X 0 1 2 3
P(x) .125 .375 .375 .125
Out of 3 coins that are tossed, what is the
probability of getting exactly 2 heads?
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Binomial Formula
Where
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• Out of 3 coins that are tossed, what is the
  probability of getting exactly 2 heads?

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The number of inaccurate gauges in a group of
four is a binomial random variable. If the
probability of a defect is 0.1, what is the
probability that only 1 is defective?
More than 1 is defective?
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Calculator

• Binomialpdf(n,p,x) this calculates the
  probability of a single binomial P(x k)
• Binomialcdf(n,p,x) this calculates the
  cumulative probabilities from P(0) to P(k)

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A genetic trait of one family manifests itself in
25 of the offspring. If eight offspring are
randomly selected, find the probability that the
trait will appear in exactly three of them. At
least 5?
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In a certain county, 30 of the voters are
Republicans. If ten voters are selected at
random, find the probability that no more than
six of them will be Republicans.
P(x lt 6) binomcdf(10,.3,6) .9894
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Binomial formulas for mean and standard deviation
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In a certain county, 30 of the voters are
Republicans. How many Republicans would you
expect in ten randomly selected voters? What is
the standard deviation for this distribution?
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Binomial Activity

• In L1 seq(x,x,0,10)
• In L2 binompdf(10, .1 ,L1)
• Sketch histogram on board

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Binomial Activity
What happened to the shape of the distribution
as the probability of success increased?
As the probability of success increases, the
shape changes from being skewed right to
symmetrical at p .5 to skewed left.
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Binomial Activity

• Calculate the mean and standard deviations for
  each of the probabilities
• What do you notice?

• As the probability of success increase,
• the means increase.
• the standard deviations increase to p .5, then
  decrease. Their values are also symmetrical.

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Geometric Distributions

• There are two mutually exclusive outcomes
• Each trial is independent of the others
• The probability of success remains constant for
  each trial.
• The random variable x is the number of trials
  UNTIL the FIRST success occurs.

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Differences between binomial geometric
distributions

• The difference between binomial and geometric
  properties is that there is NOT a fixed number of
  trials in geometric distributions!

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Other differences

• Binomial random variables start with 0 while
  geometric random variables start with 1

• Binomial distributions are finite, while
  geometric distributions are infinite

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Geometric Formulas
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Count the number of boys in a family of four
children.
Binomial X 0 1 2 3 4
Count children until first son is born
Geometric X 1 2 3 4 . . .
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What is the probability that the first son is the
fourth child born?
What is the probability that the first son is
born is at most the fourth child?
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A real estate agent shows a house to prospective
buyers. The probability that the house will be
sold to the person is 35. What is the
probability that the agent will sell the house to
the third person she shows it to?
How many prospective buyers does she expect to
show the house to before someone buys the house?
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Geometric Activity

• In L1 input numbers 1-20
• In L2 geometpdf(.1,L1)
• Sketch
• Find the means standard deviations
• What do you see?

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Geometric Activity

• Geometric distributions are skewed right and
  become more strongly skewed right as the
  probability of success increases
• Mean standard deviation of the distributions
  decrease as the probability of success increase