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Discrete Probability Distributions Random Variables and Their Probability Distributions A random variable is a real-valued function whose domain is a sample space. A random variable X is said to be discrete if it can take on only a finite number, or a countably infinite number, of possible values x. The function p(x) is called the probability function of X, provided 1. P(X = x) = p(x) 0 for all values of x 2. Example (Table 5.1) of the probability distribution of the number of closed relays of an electrical circuit. Table 5.1 Probability Distribution of Number of Closed Relays x P(x) 0 0.04 1 0.32 2 0.64 Total 1.00 Figure 5.2 Probability distribution of number of closed relays The distribution function F(b) for a random variable X is defined as F(b) = P(X b). The distribution function is also known as the cumulative distribution function (cdf). Example: The random variable X, denoting the number of relays closing properly of the electrical circuit has the probability distribution given below: P(X = 0) 0.04 P(X = 1) 0.32 P(X = 2) 0.64 Note that P(X 1.5) = P(X =0) + P(X = 1) = 0.04 + 0.32 = 0.36 The distribution function for this random variable (Figure 5.4) has the form Figure 5.3 A distribution function for a discrete random variable Expected Values of Random Variables The expected value of a discrete random variable X having probability function p(x) is given by The sum is over all values of x for which p(x) > 0. The variance of a random variable X with expected value is given by The standard deviation of a random variable X is the square root of the variance given by Example: The manager of a stockroom in a factory knows from her study of records that X, the daily demand (number of times used per day) for a certain tool, has the following probability distribution: Demand Probability 0 0.1 1 0.5 2 0.4 Find the expected daily demand for the tool and the variance. Solution: From the definition of expected value, E(X) = Therefore, the tool is used an average of 1.3 times per day. The variance is V(X) = For any random variable X and constants a and b. 1. E(aX + b) = aE(X) + b 2. V(aX + b) = a2V(X) Example: In the previous example, suppose it costs the factory $100 each time the tool is used. Find the mean and variance of the daily costs for use of this tool. Solution: Recall that X = the daily demand. Then the daily cost of using this tool is 100X. Hence E(100X) = 100 E(X) = 100(1.3) = 130 and V(100X) = 1002V(X) = 10,000(0.41) = 4,100 The factory should budget $130 per day to cover the cost of using the tool. The standard deviation of daily cost is $64.03. Example: The operators of certain machinery are required to take a proficiency test. A national testing agency gives this test nationwide. The possible scores on the test are 0, 1, 2, 3, or 4. The test brochure published by the agency gave the following information about the scores and the corresponding probabilities: Score Probability 0 0.1 1 0.2 2 0.4 3 0.2 Compute the mean score and the standard deviation. Solution: The mean score or the expected score is Since, The variance can be computed as The standard deviation of scores is and 4 0.1