VARIATION AND PROBABILITY

DISCRETE DISTRIBUTIONS

Discrete Random Variable

A discrete random variable a random variable which takes discrete values with specified probabilities.

Example - Family of 3 children.

Let X be the Random Variable (RV) = number of girls

Possible values:

        X = 3           GGG
        X = 2           GGB     GBG     BGG
        X = 1           BBG     BGB     GBB
        X = 0           BBB

Assume the 8 outcomes are equally likely so that

x	0	1	2	3
probability P(X = x)	1/8	3/8	3/8	1/8

The list of values X can take and their probabilities is called the discrete probability distribution for X.

Notation convention - use capital letters for random variables and small letters for specific values.

Example - Bernoulli trials

Each trial is an 'experiment' with exactly 2 possible outcomes, "success" and "failure" with probabilities p and 1-p.

Let X = 1 if success, 0 if failure

Probability distribution is

Results for Bernoulli trials can be simulated using MINITAB, e.g. To simulate the results of a trial of a new drug, where success (cure) has probability 0.3 for each patient, and there are 100 patients in the trial.

          RANDOM   100   C1;
          BERNOULLI   0.3.

Results are put into column 1 and look like: 0 1 0 0 0 1 0 1 . . . .

corresponding to failure, success, failure, failure, ...,(100 results)

Example - 2 dice are thrown

If the dice are fair, the 36 outcomes are all equally likely so each has probability = 1/36 .

Let X denote the total thrown. X is a random variable, and possible values of X are 2, 3, ..., 12. Each value can arise in one or more ways: for example, X=4 happens when the outcome corresponds to any of the pink cells in the table above. The probability of each particular value for X is the sum of the probabilites of its consituent outcomes;

e.g. P(X = 4) = P(1,3 or 2,2 or 3,1) = 3/36 .

The probability distribution is

x	2	3	4	. . .	10	11	12
P(X=x)	1/36	2/36	3/36	. . .	3/36	2/36	1/36