Discrete Probability Distributions

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Chapter 5

• Discrete Probability Distributions

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Roll Two Dice and Record the Sums
Physical Outcome An ordered pair of two faces
showing. We assign a numeric value to each pair
bycounting up all of the dots that show.
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A Function of Events

• Note that there may beseveral outcomes thatget
  the same value.
• This assignment of anumeric value is, in fact,a
  function.
• The domain is a set containing the
  possibleoutcomes, and the rangeis the set of
  numbersthat are assigned to theoutcomes.
• You might say, for example,f( )5.

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Random Variable

• However, we dont use f(x) notation in this case.
• This function is called a random variable and is
  typically given a capital letter name, such as X.
• Even though it is truly a function, we use it
  much the same way that we would a variable in
  algebra, except for one thing
• The random variable takes on different values
  according to a probability distribution
  associated with the underlying events.
• That is, we can never be certain what the value
  will be (random) and
• The values vary (variable) from trial to trial.

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The Probabilities of X

• We have already used the notation P(A) in
  connection with the probabilities of events.
• P is a function that relates an event to a
  probability (a number between 0 and 1).
• Similarly, we will use expressions like
  P(Xx).5, or P(X3).5.
• Why not just say P(3).5? We sometimes do, for
  short. Technically, 3 doesnt have a
  probability. Its the event, X3, that has a
  probability.
• X3 should be understood as X takes the value
  3, rather than X equals 3.
• P(X3).5 says 3 is a value of X that occurs with
  probability .5.
• Lower-case letters are used for particular
  values, upper-case for r.v. names, as in P(Xx).

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A Random Variable X for two dice
The table lists the outcomes that are mapped to
each sum, x. The n(x) column tells how many
equally likely outcomes are in each
group. P(Xx) n(x)/n(S) n(x)/36.
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Probability Histogram for X
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A Probability Function Definition

• A histogram is often called a distribution
  because it graphically depicts how the
  probability is distributed among the values.
  (Actually, a histogram is just a picture of a
  distribution, not the distribution itself.)
• We also like to have a formula that gives us the
  probability values when this is possible.
• The 2-dice toss problem gives a nice regular
  shape. Can we come up with a formula for the
  probabilities?
• It is a V-shaped function, which is typical of
  absolute value graphs. Since the vertex is at
  x7, we could try something with x-7. A little
  experimentation will lead to

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Even Easier

• Consider the toss of a single die.
• Define a random variable X as the number of spots
  that show on the top face.
• Define a probability function for X as
• Consider a coin toss. Let
• Define a probability function for X as

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Measures of Central Tendency

• Find the mean of a distribution
• Think What is a mean?
• Average of all observations
• Theoretical long run average of observations
• Calculate this from the information in the
  probability distribution

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Example of a Simple Probability Distribution

• Say we have a discrete r.v. X as follows
• Suppose we have 10 realizations of X. If the 10
  occurred in the exact long-run proportions, what
  would they be?1, 1, 1, 2, 2, 2, 2, 3, 3, 3.

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Calculate the Mean

• What then would the mean be?
• Note The mean doesnt have to be a value of X.

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Expected Value of a Discrete R.V.
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Variance of a Discrete R.V.

• Variance is also an expected value
• Standard Deviation, as always, is the square root
  of the variance.

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• Example The number of standby passengers who get
  seats on a daily commuter flight from Boston to
  New York is a random variable, X, with
  probability distribution given below. Find the
  mean, variance, and standard deviation.

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• Solution
• Using the formulas for mean, variance, and
  standard deviation
• Note 1.55 is not a value of the random variable
  (in this case). It is only what happens on
  average.

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• Example The probability distribution for a
  random variable x is given by the probability
  function
• Find the mean, variance, and standard deviation.
• Solution
• Find the probability associated with each value
  by using the probability function.

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Binary Experiments(Bernoulli Trials)

• A Bernoulli Trial is an experiment for which
  there are only two possible outcomes.
• For probability theory purposes, these are
  designated success and failure, although the
  names are arbitrary.
• Examples include a coin toss with outcomes of
  heads or tails, or any experiment where the
  results are yes or no, true or false, good or
  defective, etc.

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The Bernoulli Distribution

• A Bernoulli R.V. assigns to each outcome of a
  Bernoulli Trial a 1 for success or a 0 for
  failure.
• P(1) is denoted by p and is the parameter of the
  distribution (probability of a success).
• P(0)(1p) because 0 is the complement of 1.
• The notation q(1-p) is also used to simplify
  formulas. However, q is not another parameter,
  because its value is determined by p.

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Mean and Variance of Bernoulli
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Some Examples
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Binomial Distribution

• Suppose in a series of n Bernoulli Trials you
  keep track of the total number of successes.
• The trials are independent.
• We say p and n are the parameters of the
  distribution.
• Let X be a r.v. for the number of successes.
• Lets start with n2 and p is 1/4.
• The next slide shows the outcomes with
  corresponding values of X and probabilities.

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Binomial Probability Example

• Outcome Probability Value of X (SS)
  (1/4)(1/4)1/16 2 (SF)
  (1/4)(3/4)3/16 1 (FS)
  (3/4)(1/4)3/16 1 (FF)
  (3/4)(3/4)9/16 0
• Value of X Probability of 2 1/16 1
  6/16 0 9/16

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Mean and Variance of Binomial when n2
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Factorials and Combinations

• n-factorial is defined as follows
• A combination is read n choose r and
  represents the number of ways to choose r
  objects from n without regard to order. It can
  also be written as nCr (often on calculators).
  It is defined as follows

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Binomial Probabilities

• For a binomial r.v. X, with probability parameter
  p and n trials,
• Example Calculate the probability of 3
  successes in 5 trials if p.25.

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A Fishing Trip

• Dr. A is fond of fly-fishing in the Colorado
  mountain streams. His long-run average is one
  catch per 20 casts. On the last day of his
  vacation, He stops to eat supper. He figures he
  will cast 10 more times and then pack up. What
  is the probability he will catch one more fish?
• What is the probability he will catch at least
  one more fish?
• What is the probability he will catch more
  than one fish?

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From Binomial to Poisson

• The Binomial Distribution deals with counting
  successes in a fixed number of trials. It is
  discrete and finite-valued.
• Suppose there is no specified number of trials.
  We may want to count the number of successes
  that occur in an interval of time or space.
• Successes, or sometimes events, (not to be
  confused with probability events) refer to
  whatever we are interested in counting, such as
• the number of errors on a typed page,
• the number of flights that leave an airport in an
  hour,
• the number of people who get on the bus at each
  stop,
• or the number of flaws on the surface of a metal
  sheet.

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Poisson Distribution

• A Poisson Random Variable, X, takes on values 0,
  1, 2, 3, . . . , corresponding to the number of
  events that occur.
• Since no definite upper bound can be given, X is
  an infinitely-valued discrete random variable.
• Assumptions
• The probability that an event occurs is the same
  for each unit of time or space.
• The number of events that occur in one unit of
  time or space is independent of any others.

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Poisson Probability Function

• The probability of observing exactly x events or
  successes in a unit of time or space is given by
• .
• µ is the mean number of occurrences per unit time
  or space, that is, .
• In a most amazing coincidence, it turns out that
  !

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A Fishing Trip Too

• Dr. A is fond of fly-fishing in the Colorado
  mountain streams. His long-run average is one
  catch per hour. On the last day of his vacation,
  he stops to eat supper. He figures he will fish
  two more hours and then pack up. What is the
  probability he will catch at least one more fish?
• Note µ 2 (in a two hour period).

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More Poisson Fishing

• What is the probability Dr. A will catch one more
  fish?
• What is the probability Dr. A will catch more
  than one more fish?

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Discrete Distributions

• The examples we have just seen are common
  discrete distributions,
• That is, the r.v. in question takes only discrete
  values (with Pgt0).
• We have also seen that discrete r.v.s may be
  finite or infinite with regard to the number of
  values they can take (with Pgt0).
• There are many more such discrete distributions,
  and we should mention some of them just so you
  are aware they are available.

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• The multinomial distribution is like the
  binomial, except that it has more than two
  categories with probabilities for each. The r.v.
  is not based on a single value but a vector
  giving the counts of each possible outcome. The
  counts have to add up to the number of trials.
  You could use this, for example, to calculate the
  probability of getting 2 fives and 2 sixes in 4
  tosses of a die, written as P(0,0,0,0,2,2).

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• The geometric distribution is used when you are
  interested in the number of trials until the
  first success. This is an infinite-valued
  distribution like Poisson. You could use this,
  for example, to calculate the probability that
  you find the first defective on the 10th trial,
  if the probability of a defective is .1.
• More often, we are interested in the cumulative
  probability (finding the first defective by the
  10th trial)

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• The negative binomial distribution is an
  extension of the geometric. Instead of counting
  the number of trials until the first success, we
  count the number of failures, X, until we have r
  successes. For example, find the probability
  that you have to examine 10 items to find 3
  defectives, if the probability of a defective is
  .1.

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More on Combinations

• Combinations can be used to calculate
  probabilities for many common problems like card
  games.
• Recall the probability definition that said
  .
• If we can find the number of combinations of
  cards that fit our definition of a particular
  hand, and divide by the total number of
  combinations of cards, we can calculate
  probabilities for hands.

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Poker Hands

• First, what is the total number of hands possible
  in poker? (5-card combinations)
• This will be our denominator.

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Try a Flush

• To figure out the numerator, we can think in
  terms of the choices that have to be made to
  build the hand.
• For a flush, we first have to choose 1 suit out
  of 4. Given the suit, we have to choose 5 cards
  from the 13 in that suit.

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Not All Flushes

• However, we have to make sure we have mutually
  exclusive definitions. Not all flushes, are
  counted as flushes. Some are straight flushes
  and royal flushes.

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Final Results for Flushes

• So

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One Pair

• To get one pair (and nothing else) we need to
  select a number for the pair, then 3 other,
  different numbers for the remaining cards (order
  doesnt matter). However, that is not all. We
  must also select suits, two for the pair and 1
  for each of the other cards.

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Full House

• One more example To get a full house, you
  choose one number for the pair, then two suits
  for it, then another number for the triple, and
  three suits for it.