Chapter 4 Discrete Probability Distributions

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

A Function of Events  Note that there may be several outcomes that get the same value.  This assignment of a numeric value is, in fact, a function.  The domain is a set containing the possible outcomes, and the range is the set of numbers that are assigned to the outcomes.  You might say, for example, f( )=5.

Random Variable  However, we don’t 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.

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(X=x)=.5, or P(X=3)=.5.  Why not just say P(3)=.5? We sometimes do, for short. Technically, 3 doesn’t have a probability. It’s the event, X=3, that has a probability.  X=3 should be understood as “X takes the value 3”, rather than “X equals 3”.  P(X=3)=.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(X=x).

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(X=x) = n(x)/n(S) = n(x)/36. Outcomesxn(x) (1,1)21 (2,1),(1,2)32 (3,1),(2,2),(1,3)43 (4,1),(3,2),(2,3),(1,4)54 (5,1),(4,2),(3,3),(2.4),(1,5)65 (6,1),(5,2),(4,3),(3,4),(2,5),(1,6)76 (6,2),(5,3),(4,4),(3,5),(2,6)85 (6,3),(5,4),(4,5),(3,6)94 (6,4),(5,5),(4,6)103 (6,5),(5,6)112 (6,6)121

Probability Histogram for X

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 x=7, we could try something with |x-7|. A little experimentation will lead to

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

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

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. xP(X=x)

Calculate the Mean  What then would the mean be?  Note: The mean doesn’t have to be a value of X.

Expected Value of a Discrete R.V.

Variance of a Discrete R.V.  Variance is also an expected value  Standard Deviation, as always, is the square root of the variance.

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.

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.

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.

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.

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)=(1–p) 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.

Mean and Variance of Bernoulli

Some Examples

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.  Let’s start with n=2 and p is 1/4.  The next slide shows the outcomes with corresponding values of X and probabilities.

Binomial Probability Example  Outcome ProbabilityValue of X (SS) (1/4)(1/4)=1/16 2 (SF) (1/4)(3/4)=3/161 (FS) (3/4)(1/4)=3/161 (FF) (3/4)(3/4)=9/160  Value of X Probability of 2 1/16 1 6/16 0 9/16

Mean and Variance of Binomial when n=2

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 n C r (often on calculators). It is defined as follows:

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.

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? 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? What is the probability he will catch more than one fish?

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.

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.

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 !

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).

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?

Discrete Distributions  The examples we have just seen are common discrete distributions,  That is, the r.v. in question takes only discrete values (with P>0).  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 P>0).  There are many more such discrete distributions, and we should mention some of them just so you are aware they are available.

 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). 

 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 10 th trial, if the probability of a defective is.1.  More often, we are interested in the cumulative probability (finding the first defective by the 10 th trial):

 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.

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.”

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

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.

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.

Final Results for Flushes  So

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 doesn’t matter). However, that is not all. We must also select suits, two for the pair and 1 for each of the other cards.

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.