This section covers Discrete Random Variables, probability distribution, Cumulative Distribution Function and Probability Density Function. A probability distribution is a table of values showing the probabilities of various outcomes of an experiment. For example, if a coin is tossed three times, the number of heads obtained can be 0, 1, 2 or 3. The probabilities of each of these possibilities can be tabulated as shown:
A discrete variable is a variable which can only take a countable number of values. In this example, the number of heads can only take 4 values (0, 1, 2, 3) and so the variable is discrete. The variable is said to be random if the sum of the probabilities is one. Probability Density Function The probability density function (p.d.f.) of X (or probability mass function) is a function which allocates probabilities. Put simply, it is a function which tells you the probability of certain events occurring. The usual notation that is used is P(X = x) = something. The random variable (r.v.) X is the event that we are considering. So in the above example, X represents the number of heads that we throw. So P(X = 0) means "the probability that no heads are thrown". Here, P(X = 0) = 1/8 (the probability that we throw no heads is 1/8 ). In the above example, we could therefore have written:
Quite often, the probability density function will be given to you in terms of x. In the above example, P(X = x) = 3Cx / (2)3 (see permutations and combinations for the meaning of 3Cx). Example A die is thrown repeatedly until a 6 is obtained. Find the probability density function for the number times we throw the die. Let X be the random variable representing the number of times we throw the die. P(X = 1) = 1/6 (if we only throw the die once, we get a 6 on our first throw. The probability of this is 1/6 ). P(X = 2) = (5/6) × (1/6) (if we throw the die twice before getting a 6, we must throw something that isn't a 6 with our first throw, the probability of which is 5/6 and we must throw a 6 on our second throw, the probability of which is 1/6) etc In general, P(X = x) = (5/6)(x-1) × (1/6) Cumulative Distribution Function The cumulative distribution function (c.d.f.) of a discrete random variable X is the function F(t) which tells you the probability that X is less than or equal to t. So if X has p.d.f. P(X = x), we have: F(t) = P(X £ t) = SP(X = x). In other words, for each value that X can be which is less than or equal to t, work out the probability that X is that value and add up all such results. Example In the above example where the die is thrown repeatedly, lets work out P(X £ t) for some values of t. P(X £ 1) is the probability that the number of throws until we get a 6 is less than or equal to 1. So it is either 0 or 1. P(X = 0) = 0 and P(X = 1) = 1/6. Hence P(X £ 1) = 1/6 Similarly, P(X £ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0 + 1/6 + 5/36 = 11/36 In order to allow a broader range of more realistic problems Chapter 12 "Appendix" contains probability tables for binomial random variables for various choices of the parameters n and p. These tables are not the probability distributions that we have seen so far, but are cumulative probability distributions. In the place of the probability P(x) the table contains the probability This is illustrated in Figure 4.6 "Cumulative Probabilities". The probability entered in the table corresponds to the area of the shaded region. The reason for providing a cumulative table is that in practical problems that involve a binomial random variable typically the probability that is sought is of the form P(X≤x) or P(X≥x). The cumulative table is much easier to use for computing P(X≤x) since all the individual probabilities have already been computed and added. The one table suffices for both P(X≤x) or P(X≥x) and can be used to readily obtain probabilities of the form P(x), too, because of the following formulas. The first is just the Probability Rule for Complements.
Figure 4.6 Cumulative Probabilities
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