Lecture 16: Binomial Distribution and Continuous Distributions

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Lecture 16 Binomial Distribution and Continuous
Distributions

• Professor Aurobindo Ghosh
• E-mail ghosh_at_galton.econ.uiuc.edu

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The Binomial Distribution

• The binomial experiment can result in only one
  out of two outcomes.
• Typical cases where the binomial experiment
  applies
• A coin flipped results in heads or tails
• An election candidate wins or loses
• An employee is male or female
• A car uses 87octane gasoline, or another gasoline.

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Binomial experiment

• There are n trials (n is finite and fixed).
• Each trial can result in a success or a failure.
• The probability p of success is the same for all
  the trials.
• All the trials of the experiment are independent.
• Binomial Random Variable
• The binomial random variable counts the number of
  successes in n trials of the binomial experiment.
• By definition, this is a discrete random variable.

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• Example 6.9
• 5 of a catalytic converter production run is
  defective.
• A sample of 3 converter s is drawn. Find the
  probability distribution of the number of
  defectives.
• Solution
• A converter can be either defective or good.
• There is a fixed finite number of trials (n3)
• We assume the converter state is independent on
  one another.
• The probability of a converter being defective
  does not change from converter to converter
  (p.05).

The conditions required for the binomial
experiment are met
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• Let X be the binomial random variable indicating
  the number of defectives.
• Define a success as a converter is found to be
  defective.

X P(X) 0 .8574 1 .1354 2 .0071 3
..0001
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• Mean and variance of binomial random variable
• E(X) m np
• V(X) s2 np(1-p)
• Example 6.10
• Records show that 30 of the customers in a shoe
  store make their payments using a credit card.
• This morning 20 customers purchased shoes.
• Use Table 1 of Appendix B to answer some
  questions stated in the next slide.

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• What is the probability that at least 12
  customers used a credit card?
• This is a binomial experiment with n20 and
  p.30.

.01.. 30
0 . . 11
P(At least 12 used credit card)
P(Xgt12)1-P(Xlt11) 1-.995 .005
.995
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• What is the probability that at least 3 but not
  more than 6 customers used a credit card?

.01.. 30
0 2 . 6
P(3ltXlt6) P(X3 or 4 or 5 or 6) P(Xlt6)
-P(Xlt2) .608 - .035 .573
.035
.608
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• What is the expected number of customers who used
  a credit card?
• E(X) np 20(.30) 6
• Find the probability that exactly 14 customers
  did not use a credit card.
• Let Y be the number of customers who did not use
  a credit card.P(Y14) P(X6) P(Xlt6) -
  P(xlt5) .608 - .416 .192
• Find the probability that at least 9 customers
  did not use a credit card.
• Let Y be the number of customers who did not use
  a credit card.P(Ygt9) P(Xlt11) .995

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Continuous Probability Distributions

• A continuous random variable has an uncountably
  infinite number of values in the interval (a,b).

• The probability that a continuous variable X will
  assume any particular value is zero.

The probability of each value
1/4 1/4 1/4
1/4 1
1/3 1/3
1/3 1
1/2
1/2 1
1/2
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• To calculate probabilities we define a
  probability density function f(x).
• The density function satisfies the following
  conditions
• f(x) is non-negative,
• The total area under the curve representing f(x)
  equals 1.
• The probability that x falls between a and b is
  found by calculating the area under the graph of
  f(x) between a and b.

Whole Area 1
P(altxltb)
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Uniform Distribution

• A random variable X is said to be uniformly
  distributed if its density function is

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• Example 7.1
• The time elapses between the placement of an
  order and the delivery time is uniformly
  distributed between 100 and 180 minutes.
• Define the graph and the density function.
• What proportion of orders takes between 2 and 2.5
  hours to deliver?

f(x) 1/80 100ltxlt180
P(120lt xlt150) (150-120)(1/80) .375
1/80
x
100
180
120
150