The elapsed time (in minutes) between the arrival of west-bound trains

at the St. George subway station is an exponential random variable with a

value of ? = .2.

a) What are the expected value and variance of X?

b) What is the probability that 10 or more minutes will elapse between

consecutive west-bound trains?

c) What is the probability that 10 or more minutes will elapse between

trains, given that at least 8 minutes have already passed since the previous train arrived? Hint: What proportion of the probability weight

that remains, given that a waiting time of less than 8 minutes is no

longer possible, lies in the interval 8 minutes to 10 minutes?

17. The number of houses sold each month by a top real estate agent is a

Poisson random variable X with ? = 4.

a) What are the expected value and standard deviation of X

John Daly is among the best putters on the PGA golf tour. He sinks 10

percent of all puts that are of length 20 feet or more. In a typical round of

golf, John will face puts of 20 feet or longer 9 times. What is the probability

that John will sink 2 or more of these 9 puts? What is the probability that

he will sink 2 or more, given that he sinks one? Hint: If we know he is going

to sink at least one then the only remaining possibilities are that he will sink

only that one or two or more. What fraction of the remaining probability

weight (excluding the now impossible event that he sinks zero) falls on the

event 'two or more'.

12. Let X and Y be two random variables. Derive formulae for E{X + Y },

E{X ? Y }, ?

2{X + Y }, and ?

2{X ? Y }. Under what conditions does

?

2{X + Y } = ?

2{X ? Y }?

13. According to the Internal Revenue Service (IRS), the chances of your

tax return being audited are about 6 in 1000 if your income is less than

$50,000, 10 in 1000 if your income is between $50,000 and $99,999, and 49

in 1000 if your income is $100,000 or more (Statistical Abstract of the United

States: 1995.

a) What is the probability that a taxpayer with income less than $50,000

will be audited by the IRS? With income between $50,000 and $99,999?

With income of $100,000 or more?

b) If we randomly pick five taxpayers with incomes under $50,000, what

is the probability that one of them will be audited? That more than

one will be audited? Hint: What are n and p here?

Business Statistics Question 10 Not yet answered Points out of 2.00 P Flag question The weekly income distribution for production supervisors in a local manufacturing plant is normally distributed with a mean weekly income of $1,000 and a standard deviation of $100. What would a production supervisor's weekly income need to be in order for him/her to earn in the top 5%? Show your answer to two decimal places (for example, 925.45). Answer: Previous page Next pageProb. 4 The lifetime of a certain component, in years, has the probability density function {PDF} _ a\": 1' :_e I] am lo, 1 i .1 Two such components; whose lifetimes are independent, are available. As soon as the first component fails, it is replaced with the second component. Let H be a random valiable that denotes the lifetime of the first compo nent, and let F denote the lifetime of the second component {:1} Find the joint cumulative distribution function [CDF] of A" and F. {b} Find the expected value of the sum, i.e., EUL" + 1'). {c} Find PL? 5 1|? .2: 1). {:1} Find For + Y 1: a). 6) [5 points each] A drawer contains 3 red, 2 green, and 4 blue socks. Two socks are selected at random. Let X denote the number of red socks, and Y denote number of green socks selected. The following table is prepared in order to display the joint probability mass function of X and Y. 0 2 a. Show that P(X = 0. Y = 0) = ] and P(X - 1, Y = 0) - |. Give all the details for full credit. Then complete the table, b. Let F denote the joint cumulative distribution function (ed f.) of X and Y. Compute F(1,0).Suppose that two discrete random variable X and Y have the following joint probability mass function. X Y -1 0 -1 1/16 1/8 1/16 0 1/8 1 1/8 1/8 1/16 1/16 (a) Write the px(x), the marginal probability mass function of X. (b) Write the py(y), the marginal probability mass function of Y. (c) Are X and Y independent? Why or why not? (d) What is F(0.5, -0.3), the joint cumulative distribution function (CDF) evaluated at x = 0.5 and y = -0.3