statistics

Statistical Inference I: J. Lee Assignment 8 Due on Dec 1, 2015

Problem 1. Let X have probability density function

f(x) =

 

x/2 if 0 < x < 1 1 if 2.25 < x < 3 0 otherwise

Compute the moment generating function, MX (t) of X.

Problem 2. Consider a random variable X whose cumulative distribution function is given by

F (x) =

 

0 if x < −2 0.2 if − 2 ≤ x < 0 0.5 if 0 ≤ x < 2.2 0.6 if 2.2 ≤ x < 3 0.6 + q if 3 ≤ x < 4 0.6 + 2q if 4 ≤ x < 5.5 1 if x ≥ 5.5

We are also told that P (X > 3.3) = 0.25.

(a) Compute the moment generating function, MX (t) of X.

(b) Use the moment generating function to compute E(X) and E(X2), and verify your answers by com- puting the values directly.

Problem 3. Suppose that the average number of parking tickets given in front of LeBow (Market Street) is three per day.

(a) Estimate the probability, p, that at least five parking tickets will be given out in front of LeBow (Market Street) tomorrow. (What inequality are you using?)

(b) Assume now (for parts (b), (c), and (d)) that you are told that the variance of the number of tickets in any one day is 9. Now give an estimate of p that takes advantage of knowing the variance (using an inequality).

(c) Give a Central Limit Theorem estimate for the probability q that in the month of December (which has 31 days, and we consider each day to be like any other day) there are more than 75 parking tickets given out.

(d) Use an inequality to get the best bounds you can on the probability q estimated in part (c).

Problem 4. I measure the length of my bed 50 times, each time making a random error, uniformly distributed over (-1,1) centimeters. What is the probability that the average of these 50 measurements differs from the actual value by less than 0.25 centimeters?

Problem 5. The average number of thefts at LeBow is three per month.

(a) Estimate the probability, p, that at least six thefts occur at LeBow during December. (What inequality are you using?)

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(b) Assume now (for parts (b), (c), and (d)) that you are told that the variance of the number of thefts at LeBow in any one month is 2. Now give an improved estimate of p (using an inequality).

(c) Give a Central Limit Theorem estimate for the probability q that during the next 5 years (12 months per year) there are more than 150 thefts at LeBow.

(d) Use an inequality to get the best bounds you can on the probability q estimated in part (c).

Problem 6. Starbucks manager at LeBow wants to find:

(a) the probability of the total average cost less than or equal to $1000 per day. He is only concerned about the cost of waiting time of customers. Customers spend time at the checkout register with mean of 4 minutes and standard deviation of 4 minutes. On average, he got 100 customers per day. 1 minute of waiting time is considered as cost of $0.5.

(b) the probability of the total expected revenue greater than or equal to $1500 per day. On average, each customer spends $4 at a time with standard deviation of $1.

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