1. Biostatistics 410.645.01 Class 3 Discrete Probability Distributions 2/8/2000

2. Probability distributions of discrete variables A table, graph, formula, or other device used to specify all possible values of a discrete random variable along with their respective probabilities –P(X=x) Tables – value, frequency, probability Graph – usually bar chart or histogram Formula - Binomial distribution

3. Cumulative Distributions Probability that X is less than or equal to a specified value, x I Calculated by adding successive probabilities P(X=x i ) Easier to work with for many applications P(X  x i ) Theoretical distribution can be compared to sample distribution to determine appropriateness of theoretical distribution

4. Theoretical Probability Distributions Why bother? Isn’t observation enough? –If we know (reasonably) that data are from a certain distribution, than we know a lot about it Means, standard deviations, other measures of dispersion –That knowledge makes it easier to make statistical inference; i.e., to test differences Many types of distributions –1300+ have been documented in the literature Three main ones –Binomial (discrete - 0,1) –Poisson (discrete counts) –Normal (continuous)

5. Binomial Distribution Derived from a series of binary outcomes called a Bernoulli trial When a random process or experiment, called a trial, can result in only one of two mutually exclusive outcomes, such as dead or alive, sick or well, the trial is called a Bernoulli trial

6. Bernoulli Process A sequence of Bernoulli trials forms a Bernoulli process under the following conditions –Each trial results in one of two possible, mutually exclusive, outcomes: “success” and “failure” –Probability of success, p, remains constant from trial to trial. Probability of failure is q = 1-p. –Trials are independent; that is, success in one trial does not influence the probability of success in a subsequent trial.

7. Bernoulli Process - Example Probability of a certain sequence of binary outcomes (Bernoulli trials) is a function of p and q. For example, a particular sequence of 3 “successes” and 2 “failures” can be represented by p*p*p*q*q; = p 3 q 2 However, if we ask for the probability of 3 “successes” and 2 “failures” in a set of 5 trials, then we need to know how may possible combinations of 3 successes and 2 failures out of all of the possible outcomes there are.

8. Combinations Based on last example, it is clear that we need to calculate more easily the probability of a particular result –If a set consists of n objects, and we wish to form a subset of x objects from these n objects, without regard to order of the objects in the subset, the result is called a combination The number of combinations of n objects taken x at a time is given by – n C k = n! / (k! ( n-k)!) –Where k! (factorial) is the product of all numbers from k to 0 0! = 1

9. Combinations From this, we can determine the binomial probability density function –f(x) = n C x p x q n-x for x=0,1,2,3…,n – = 0 elsewhere –This is called the binomial distribution

10. Permutations Similar to combinations –If a set consists of n objects, and we wish to form a subset of x objects from these n objects, taking into account the order of the objects in the subset, the result is called a permutation The number of permutations of n objects taken x at a time is given by – n P k = n! / ( n-k)!

11. Binomial Table Normally, we would look up probabilities in the Binomial Table (Table 1 in the Appendix) –Tables the Binomial probability distribution function –P (X=k) –Find probability that x=4 successes when n trials = 10 and p of success = 0.3 –Find probability that x  4 –Find probability that x  5

12. Binomial Table when p > 0.5 Restate problem in terms of failures –P(X=k|n, p>0.50) = P(X=n-k|n,1-p) –Treat p = q for purposes of using the table –For cumulative probabilities: P(X  k|n,p>0.5) = P(X  n-xk|n,1-p) –For the probability of X  some k when p > 0.5, P(X  k | n, p>0.5) = P(X  n-k|n,1- p)

13. Binomial parameters Mean –  = np Variance –  2 = np(1-p) Appropriateness in sampling situations –Appropriate if n small relative to N –Otherwise, not really in a sampling situation

14. Poisson Distribution Used for counting processes If k is the number of occurrences of some random event in an interval of time or space, the probability that k will occur is given by –where µ is the average number of occurrences of the random event ( ) in the interval t. –e = 2.7183 Parameters of the Poisson distribution –Mean = –Variance =

15. Poisson Process Assumptions –Occurrences of events are independent; i.e., occurrence of an event has no effect on the probability of the occurrence of a second event –Theoretically, an infinite number of occurrences of the event must be possible in the interval –Probability of the single occurrence of the event in a given interval is proportional to the length of the interval; i.e., constant event rate –In an infinitesimally small portion of the interval, the probability of more than one occurrence of the event in negligible; i.e., the event times are unique and discrete

16. Application of the Poisson Distribution Cancer recurrences –Bladder cancer –Breast cancer Infections Earthquakes Plane crashes

17. Using Table of Poisson Distribution Use Table 2 to look up probabilities for Poisson variables –Tables exact Poisson probabilities Pr(X=k) –Example Probability of obtaining exactly 4 events for a Poisson distribution with  = 6.0 Probability of at least 12 Probability of 3 or less

18. Poisson Approximation to the Binomial Distribution When n is large and p is small, the Poisson is a reasonable approximation to the Binomial –Poisson is easier to work with