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05 STAT HWA q discrete rv

# 05 STAT HWA q discrete rv - 1 Dr Valerie R Bencivenga...

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1 Dr. Valerie R. Bencivenga Economics 329 PRACTICE HOMEWORK #5A: DISCRETE RANDOM VARIABLES 1. Each day Alpha Bikeshop sells bicycles according to the following probability distribution: X = daily number of bikes sold x 0 1 2 3 4 P x (x) .1 .3 .3 .2 .1 The numbers of bikes sold on different days are statistically independent. a. What are the mean and variance of number of bikes sold? b. Su ppose that the bike salesperson’ s daily income is \$30, plus \$20 for each bike sold that day. What are the mean and variance of the salesperson’ s daily income? c. What are the mean and variance of income over a two-day period (total income)? How do they compare with your answers to part b? d. Consider average income over a two-day period. What are the mean and variance of average income? How do they compare with your answers to part b? 2. Which of the following are probability functions? Explain. a. otherwise 0 4 , 3 , 2 , 1 x , x ) 10 / 1 ( ) x ( P i i i X b. otherwise 0 10 ,..., 3 , 2 , 1 x , 10 / 1 ) x ( P i i X c. otherwise 0 ... , 3 , 2 , 1 x , 2 . ) x ( P i x i X i 3. This problem continues from problem 2 of Assignment #3. a. Define a random variable X = number of hits in three attempts. Give the probability distribution of X. b. Does ) win ( P 2 X P ? Explain. (Recall you calculated win P in Practice Homework #4a, problem 2.) c. Is X a binomial random variable? Explain. 4. An insurance company is interested in the joint probability distribution of X = number of cars and Y = number of drivers. The table below gives the relative frequencies obtained from a large random sample of households. These relative frequencies may be interpreted as probabilities. a. What is the probability that a randomly-chosen household has two cars and one driver? b. What is the expected number of drivers? What is the variance of number of drivers?

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