Discrete Random Variables:

1

• Discrete Random Variables
• The Binomial Distribution

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Bernoullis trials

• J. Bernoulli (1654-1705) analyzed the idea of
  repeated independent trials for discrete random
  variables that had two possible outcomes success
  or failure
• In his notation he wrote that the probability of
  success is denoted by p and the probability of
  failure is denoted by q or 1-p

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Binomial distribution

• The binomial distribution is just n independent
  individual (Bernoulli) trials added up.
• It is the number of successes in n trials.
• The sum of the probabilities of all the
  independent trials totals 1.
• We can define a success as a 1, and a failure
  as a 0.

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Binomial distribution

• x is a binomial distribution if its probability
  function is
• Examples (note success/failure could be
  switched!)

probability of success
probability of failure
Situation x1 (success) x0 (failure)
Coin toss to get heads Turns up heads Turns up tails
Rolling dice to get 1 Lands on 1 Lands on anything but 1
While testing a product, how many are found defective Product is defective Product is not defective
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Binomial distribution

• The binomial distribution is just n independent
  Bernoulli trials added up

• It requires that the trials be done with
  replacement ex. testing bulbs for defects

• Lets say you make many light bulbs
• Pick one at random, test for defect, put it back
• Repeat several times
• If there are many light bulbs, you do not have to
  replace (it wont make a significant difference)
• The result will be the binomial probability of a
  defective bulb (defective total sample).

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Binomial distribution - formula

• Lets figure out a binomial random variables
  probability function or formula
• Suppose we are looking at a binomial with n3
  (ex. 3 coin flips heads is a success)
• We will start with all tails P(x0)
• Can happen only one way 000
• Which is (1-p)(1-p)(1-p)
• Simplified (1-p)3

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Binomial distribution - formula

• Lets figure out a binomial probability function
  (for n 3)
• This time we want 1 success plus 2 failures (ex.
  1 heads 2 tails, or P(x1))
• This can happen three ways 100, 010, 001
• Which is p(1-p)(1-p)(1-p)p(1-p)(1-p)(1-p)p
• Simplified 3p(1-p)2

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Binomial distribution - formula

• Lets figure out a binomial probability function
  (for n 3)
• We want 2 successes P(x2)
• Can happen three ways 110, 011, 101, or
• pp(1-p)(1-p)ppp(1-p)p, which simplifies to..
• 3p2(1-p)

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Binomial distribution - formula

• Lets figure out a binomial probability function
  (n 3)
• We want all 3 successes P(x3)
• This can happen only one way 111
• Which we represent as ppp
• Which simplifies to p3

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Binomial distribution - formula

• Lets figure out a binomial probability function
  in summary, for n 3, we have
• P(x)
• (Where x is the number of successes ex. of
  heads)

The sum of these expressions is the binomial
distribution for n3. The resulting equation is
an example of the Binomial Theorem.
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Binomial distribution - formula

• A quick review of the Binomial Theorem
• If we use q for (1 p), then

• p3 3p2q 3pq2 q3 (p q)3

• which is an example of the formula
• (a b)n ____________________
• (if you forget it, check it in your text)

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Binomial distribution - formula

• Lets figure out a binomial r.v.s probability
  function (the quick way to compute the sum of the
  terms on the previous slide) - now heres the
  formula
• In general, for a binomial

(the of x successes with probability p in n
trials)
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Binomial distribution - formula

• Which can be written as

or P(x) nCx px(1 p)n-x

• This formula is often called the general
  term of the binomial distribution.

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Expected Value

• The expected value of a binomial distribu-tion
  equals the probability of success (p) for n
  trials

• E(X) also equals the sum of the probabilities in
  the binomial distribution.

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Binomial distribution - Graph

• Typical shape of a binomial distribution
• Symmetric, with total P(x) 1

Note this is a theoretical graph how would an
experimental one be different?
P
x
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Binomial distribution - example

• A realtor claims that he closes the deal on a
  house sale 40 of the time.
• This month, he closed 1 out of 10 deals.
• How likely is his claim of 40 if he only
  completed 1/10 of his deals this month?

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Binomial distribution - example

• By using the binomial distribution function, its
  possible to check if his assessment of his
  abilities (i.e. 40 closes) is likely

P(0 deals)
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Binomial distribution - example

• So it seems pretty unlikely that his assess-ment
  of his abilities is right
• The probability of closing 1 or fewer deals out
  of 10 if (as he claims) he closes deals 40 of
  the time is less than 5 or less than 1/20.
• What of closes do you think would have the
  highest probability in this distribution, if his
  claim was right?

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Binomial distribution - example

• Now see if you can determine the expected number
  of closings if he had 12 deals this month,
  assuming 40 success.

• We need the values of n ( ___) and of p (
  ____).

• E(X) np ______ - this means that we would
  expect him to close about _____ deals, if his
  claim is correct. End of first example.

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Binomial Distribution ex. 2

• Alex Rios has a batting average of 0.310 for the
  season. In last nights game, he had 4 at bats.
  What are the chances he had 2 hits?
• You try this one! First ask 3 questions

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Binomial Distribution ex. 2

• Is getting a hit a discrete random variable?
• Is this a Bernoulli trial? How would you
• define a success and a failure?

• Is each time at bat an independent event?

• If you can answer yes to the three questions
  above, then you can use the binomial distribution
  formula to answer the problem.

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Binomial Distribution ex. 2

• First determine the following values
• The number of trials (Alex is at bat __ times)
  this is the value of n
• The probability of success (Alexs average is
  ___) this is p
• The probability of failure 1 p ___
• The of successes asked for (his chances of
  getting ___ hits) this is x

• Now you can use the formula

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Binomial Distribution ex. 2

• Put in the values from the previous screen, and
  discuss your answers. pause here

• Did you get P(2) 0.275?
• Is 2 the most likely number of hits for Alex
  last night? How about 1 or 3?

• P(0 or 1 or 2 or 3 or 4 hits) _____?

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Hypergeometric distribution

• What happens if you have a situation in which the
  trials are not independent (this most often
  happens due to not replacing a selected item).
• Each trial must result in success or failure, but
  the probability of success changes with each
  trial.

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Hypergeometric distribution

• Consider taking a sample from a population, and
  testing each member of the sample for defects.
• Do this sampling without replacement.

• As long as the sample is small compared to the
  population, this is close to binomial.
• But if the sample is large compared to the
  population, this is a hypergeometric dist.

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Hypergeometric dist. - formula

• A hypergeometric distribution differs from
  binomial ones since it has dependent trials.
• Probability of x successes in r dependent trials,
  with number of successes a out of a total of n
  possible outcomes

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Hypergeometric dist. - formula

• The full version of this formula is

• Expected Value the average probability of a
  success is the ratio of success overall (a/n)
  times r trials