DISCRETE PROBILITY DISTRIBUTIONS

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DISCRETE PROBILITY DISTRIBUTIONS
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Random Variables
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Some Definitions

• A random variable is a variable (typically
  represented by x) that has a single numerical
  value, determined by chance, for each outcome of
  a procedure.
• A probability distribution is a graph, table, or
  formula that gives the probability for each value
  of the random variable.

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More Definitions

• A discrete random variable has either a finite
  number of values or a countable number of values,
  where countable refers to the fact that there
  might be infinitely many values, but they can be
  associated with a counting process.
• A continuous random variable has infinitely many
  values, and those values can be associated with
  measurements on a continuous scale without gaps
  or interruptions.

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Example

• Consider a couple that wants children. Suppose
  they want to have three children. Let X be the
  number of girls the couple has. Give the
  probability distribution of X.

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Requirements for a Probability Distribution

• where x assumes all possible
  values.
• for every individual value of
  x.

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Example

• Recall our couple that wants to have three
  children. Let the random variable X represent the
  number of girls out of the three children.
• Give the probability distribution for X.

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Example (continued)

• Graph the probability distribution for X.

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Mean, Variance, and Standard Deviation of a
Probability Distribution

• Given a probability distribution for the random
  variable X

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Round-off Rule for , , and

• Round results by carrying one more decimal place
  than the number of decimal places used for the
  random variable X. If the values of X are
  integers, round , , and to one
  decimal place.

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Example

• Calculate the mean, variance, and standard
  deviation for the random variable X from the last
  example.

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Using Probabilities to Determine When Results Are
Unusual

• Unusually high number of successes x successes
  among n trials is an unusually high number of
  successes if
• Unusually low number of successes x successes
  among n trials is an unusually low number of
  successes if

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Expected Value

• The expected value of a discrete variable is
  denoted by E, and it represents the average value
  of the outcomes. If is obtained by finding the
  value of

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Example

• When you give a casino 5 for a bet on the pass
  line in a casino game of dice, there is a
  251/495 probability that you will lose 5 and
  there is a 244/495 probability that you will make
  a net gain of 5. (If you win, the casino gives
  you 5 and you get to keep your 5 bet, so the
  net gain is 5.) What is your expected value? In
  the long run, how much do you lose for each
  dollar bet?

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Example

• Sara is a 60-year-old Anglo female in reasonably
  good health. She wants to take out a 50,000 term
  (that is, straight death benefit) life insurance
  policy until she is 65. The policy will expire on
  her 65th birthday. The probability of death in a
  given year is provided by the Vital Statistics
  Section of the Statistical Abstract of the United
  States (116th Edition). Sara is applying to
  the Mutual of Burbank Insurance Company for her
  term insurance policy.

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Example (continued)

• What is the probability that Sara will die in her
  60th year? Using this probability and the 50,000
  death benefit, what is the expected loss to
  Mutual of Burbank Insurance?
• Repeat part a for years 61, 62, 63, and 64. What
  would be the total expected loss to Mutual of
  Burbank Insurance of the years 60 to 64?
• If Mutual of Burbank wants to make a profit of
  700 above the expected total loss paid out for
  Saras death, how much should it charge for the
  policy?

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Binomial Probability Distributions
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Binomial Probability Distribution

• A binomial probability distribution results from
  a procedure that meets all the following
  requirements
• The procedure has a fixed number of trials.
• The trials must be independent.
• Each trial must have all outcomes classified into
  two categories.
• The probabilities must remain constant for each
  trial.

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Notation for Binomial Probability Distributions

• S and F (success and failure) denote the two
  possible categories of all outcomes p and q will
  denote the probabilities of S and F,
  respectively, so P(S) p
  (p probability of a success) P(F) 1
  p q (q probability of a
  failure)n denotes the fixed number of
  trials.x denotes a specific number of
  successes in n trials, so x can be any
  whole number between 0 and n, inclusive.p
  denotes the probability of success in one of the
  n trials.q denotes the probability of failure
  in one of the n trials.P(x) denotes the
  probability of getting exactly x successes among
  the n trials.

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Binomial Probability Formula

• In a binomial distribution, probabilities can be
  calculated by using the binomial probability
  formulafor x 0, 1, 2, . . ., nwhere n
  number of trials x number of
  success among n trials p
  probability of success in any one trial
  q probability of failure in any one trial
  (q 1 p)

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Example

• Consider our couple that wants to have six
  children.
• What is the probability that exactly three of the
  six children are girls?
• What is the probability that exactly four of the
  children are girls?

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Calculating Binomial Probabilities

• Using the binomial probability formula.
• Using Table A-1 in Appendix A.
• Using Technology (TI 83 and TI 84)
• P(X x) binompdf(n, p, x)
• P(X x) binomcdf(n, p, x)

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Independence and Large Populations

• When sampling without replacement, the events can
  be treated as if they were independent if the
  sample size is no more than 5 of the population
  size. (That is, .)

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Example

• Suppose that 90 of all registered California
  voters favor banning the release of information
  from exit polls in presidential elections until
  after the polls in California close. A random
  sample of 25 California voters is selected.
• What is the probability at most 20 favor the ban?
• What is the probability that at least 20 favor
  the ban?

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Rationale for Binomial Probability Formula

• Since there are only two outcomes (S and F),
  then
• there are permutations of n elements
  consisting of x Ss andn x Fs, and
• and the probability of getting x Ss followed by
  n x Fs is
• Thus the probability of x success out of n
  trials is

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Mean, Variance, and Standard Deviation for the
Binomial Distribution
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Binomial Distribution Formulas

• For Binomial Distributions

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Example

• Recall our couple that wants to have six
  children. Let the random variable X be the number
  of girls. Calculate the mean, variance, and
  standard deviation for the number of girls in the
  family.

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Example

• Suppose that 90 of all registered California
  voters favor banning the release of information
  from exit polls in presidential elections until
  after the polls in California close. A random
  sample of 25 California voters is selected.
• Calculate the mean, variance, and standard
  deviation for the number who favor the ban.

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The Poisson Distribution
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Poisson Distribution

• The Poisson distribution is a discrete
  probability distributions that applies to
  occurrences of some event over a specified
  interval. The random variable X is the number of
  occurrences of the event in an interval. The
  interval can be time, distance, area, volume, or
  some similar unit. The probability of the event
  occurring x times over an interval is given
  bywhere

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Requirements for the Poisson Distribution

• The random variable X is the number of
  occurrences of an event over some time interval.
• The occurrences must be random.
• The occurrences must be independent of each
  other.
• The occurrences must be uniformly distributed
  over the interval being used.

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Parameters of the Poisson Distribution

• The Poisson Distribution has these parameters
• The mean is
• The standard deviation is

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Poisson vs. Binomial

• The binomial distribution is affected by the
  sample size n and the probability p, whereas the
  Poisson distribution is affected only by the mean
  .
• In a binomial distribution, the possible values
  of the random variable X are 0, 1, . . ., n, but
  a Poisson distribution has possible X values of
  0, 1, 2, . . . with no upper limit.

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Using Technology (TI-83 and TI-84)

• P(X x) poissonpdf( , x)
• P(X x) poissoncdf( , x)

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Example

• A large proportion of small businesses in the
  United States fail during the first few years of
  operation. On average, 1.6 businesses file for
  bankruptcy per day in a large city.
• Find the probability that exactly three
  businesses will file for bankruptcy on a given
  day in this city.
• Find the probability that less than three
  businesses will file for bankruptcy on a given
  day in this city.