Means and Variances of Random Variables - PDF document

HW 3 Means and Variances of Random Var
HW 3 Means and Variances of Random Variables Statistics  Suppose that X is a discrete random variable whose distribution is Value of X: x1 x2 x3 … xk Probablily: p1 p2 p3 … pk  To find the mean (also called the expected value) of X, multiply each possible value by its probability, then add all the products: µx = x1p1 + x2p2 + … + xkpk = ∑ xipi 1. Example 7.1 gives the distribution of grades (A = 4, B = 3, and so on) in a large class as Find the average (that is, the mean) grade in this course.  Suppose that X is a discrete random variable whose distribution is Value of X: x1 x2 x3 … xk Probablily: p1 p2 p3 … pk  And that µ is the mean of X.  The variance of X is σ2x = (x1 - µx)2p1 + (x2 - µx)2p2 + … + (xk - µx)2pk = ∑ (xi - µx)2

pi  The standard deviation
pi  The standard deviation σx of X is the square root of the variance. Find the standard deviation σx of the distribution of grades. 2. How do rented housing units differ from units occupied by their owners? Find the mean number of rooms for both types of housing unit. 3. Keno is a favorite game in casinos, and similar games are popular with the states that operate lotteries. Balls numbered 1 to 80 are tumbled in a machine as the bets are placed, then 20 of the balls are chosen at random. Players select numbers by marking a card. The simplest of the many wagers available is “Mark 1 Number.” Your payoff is $3 on a $1 bet if the number you select is one of those chosen. Because 20 of 80 numbers are chosen, your probability of winning is 20/80, or 0.25. (a) What is the probability distribution (the outcomes and their probabilities) of the payoff X on a single day? (b) What is th

e mean payoff μx? (c) In the
e mean payoff μx? (c) In the long run, how much does the casino keep from each dollar bet? 4. In an experiment on the behavior of young children, each subject is placed in an area with five tows. The response of interest is the number of toys that the child plays with. Past experiments with many subjects have shown that the probability distribution of the number X of toys played with is as follows: Number of toys x1: 0 1 2 3 4 5 Probability pi: 0.03 0.16 0.30 0.23 0.17 0.11 (a) Calculate the mean. (b) Calculate the standard deviation. 5. Below is a distribution for number of visits to a dentist in one year. X = # of visits to the dentist. a. Determine the expected value, variance and standard deviation. 6. Now that the new models are here, a car dealership has lowered prices on last year’s models. An aggressive salesp

erson estimates the following probabilit
erson estimates the following probability distribution of X, the number of cars that she’ll sell next week. a. Determine the expected value, variance and standard deviation. b. Suppose that this salesperson earns a $200 commission for each car sold. What are her expected wages for next week? What is the standard deviation of those wages? c. Suppose the employee’s complain to the owner about their pay. They want some type of guaranteed salary and more commission. The owner relents and decides they get $150 each plus $250 commission for each car sold. What are her expected wages for next week? What is the standard deviation of those wages? 7. A club sells raffle tickets for $5 each. There are 10 prizes of $25 and one price of $100. If 200 tickets are sold, determine the probability distribution. What are your expected winnings per ticket? Have you paid too much for the ticket (duh)? Expla