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Random Variables A random variable, X, represents a numerical value assigned to a possible outcome of a probability experiment. Random variables are often used to describe events. Example: Consider tossing a coin three times. The sample space S for this experiment is S = { HTT, HTT, THT, HHT, HTH, THH, HHH, TTT} Assign to each outcome the number, X, the number of heads that occur in that outcome. Outcome HTT HTT THT HHT HTH THH HHH TTT # of Heads 1 1 1 2 2 2 3 0 That means the random variable X, takes the values, 0,1, 2, and 3. Notation: P ( X x ) ---------- means the probability that X takes on the value x P (a X b) -------means the probability that X takes on the values between a and b both inclusive. Evaluate the following probabilities: 3 1. P ( X 2) 8 2. P ( X 2) P ( X 0) P ( X 1) P ( X 2) 3. P ( X 2) P ( X 0) P ( X 1) 1 3 3 7 8 8 8 8 1 3 4 1 8 8 8 2 If the random variable, X can take only the distinct values x1, x 2 , x3, ... x n . Then the expected value of X is the sum of the distinct values, weighted by their respective probabilities. The expected value of X is also called the mean of X, and is denoted by x. n Thus we have x E ( X ) xi P( X xi ) i 1 Expected Value State run monthly lottery can sell 100,000tickets at $2 a piece. A ticket wins $1,000,000with a probability 0.0000005, $100 with probability 0.008 and $10 with probability 0.01. On an average how much can the state expect to profit from the lottery per month? 1. Consider the lottery described in Example given above. How much income per month could the state expect to average if it lowered the probability of winning $1,000,000 to 0.0000004? 2. Consider the lottery describe in Example . The state plans to keep the same probabilities, but raise the top prize above $1,000,000. How high could the prize go and still give the state an expected value of $0.40 per ticket? 3.If you invest in a new restaurant there is a 35% chance that you lose $90,000, a 45% chance that you break even, and a 20% chance that you make $170,000. Compute the expected value of the investment. 4. You can invest in either Project A or Project B. If you invest in Project A, there is a 30% chance that you lose $26,000, a 50% chance that you break even, and a 20% chance that you make $68,000. If you invest in Project B, there is a 20% chance that you lose $71,000, a 65% chance that you break even, and a 15% chance that you make $143,000. Based on the expected value of each, which investment should you make? 5.As part of a sales promotion you will let the first 100 customers pick a gift certificate from a box of 1,000 certificates, some of which are for $1 and some of which are for $100. After a draw, the value of the certificate is noted and the piece of paper is replaced in the box. Thus, each customer has the same chance of getting $100 as any other customer. What is the largest number of $100 certificates that you can put into the box if the entire promotion is to have an expected cost of at most $300? Start: Let n be the number of $100 certificates in the box and let C be the random variable that gives the value of one certificate which is drawn at random. Since there will be 100 draws, the expected cost of the promotion is 100 times the expected value of each customer’s draw.