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Name: ___________________ Random Variable Homework #1 A small store keeps track of the number X of customers that make a purchase during the first hour that the store is open each day. Based on the records, X has the following probability distribution. X = customers who make purchase during the first hour a store is open P(X) 0 1 2 3 4 0.1 0.1 0.2 0.4 0.2 1. What is the probability that no customers make a purchase during the first hour that the store is open? 2. What is the probability that the number of customers that make a purchase during the first hour that the store is open is more than 1 but no greater than 3? 3. Find the mean number of customers that make a purchase during the first hour that the store is open. 4. Find the standard deviation of the number of customers that make a purchase during the first hour that the store is open. 5. Let the random variable C represent the daily profit made from selling clothing and S represent daily profit made from selling shoes on a randomly selected day by a certain store. C and S are normally distributed with a means of $360 and $250 and standard deviations $50 and $45, respectively. What is the mean and standard deviation of a random variable T that is the total profit of selling clothing and shoes (T = C + S)? 6. What is the probability that the total profit (T) of a randomly selected day will be greater than $650? A die is loaded in such a way that the probabilities of getting 1, 2, 3, 4, 5, and 6 are 1/2, 1/6, 1/12, 1/12, 1/12 and 1/12, respectively. 7. Construct a probability distribution for the data, and draw a probability distribution histogram. 8. Find P(X > 3.5). 9. Find P(2 ≤ X < 4.2). 10. Find the smallest number A for which P(X < A) > 0.6. Chapter 7 A certain probability density function is shown below. 11. Is this a discrete or continuous random variable and explain your answer 12. Is this a valid probability density function and explain your answer. 13. Find P(X < 3). 0.4 0.3 0.2 0.1 0 1 2 3 4 5 14. Find P(2 < X < 4). 15. Find P(2 ≤ X ≤ 4). 16. Find P(X = 3) The probabilities that a customer selects 1, 2, 3, 4, or 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively. 17. Construct a probability distribution (table) for the data, and verify that this is a legitimate probability distribution. 18. Find the mean of the random variable, X. 19. Find the standard deviation of X. 20. Write both the expression (in terms of X) and find the probability of a customer selecting no more than 2 items? 21. Write both the expression (in terms of X) and find the probability of a customer selecting at least 3 items? Chapter 7