DSO 516 Probability and Data Modeling Homework #2 complete solutions correct answers key

DSO 516 Probability and Data Modeling Homework #2 complete solutions correct answers key

Professor Amy R. Ward

Question 1. (20 points; 10 per part) You roll a 6-sided dice. Let the random variable X represent the

outcome of the die roll. Define the function f(z):=z2.

a. Calculate f(E[X]).

b. Calculate E[f(X)].

Question 2. (20 points; 10 per part) You live a 10 minute walk from the Culver City Expo line stop, which

takes you directly to campus. The time on the train is 20 minutes. The train departs every 10 minutes,

and your arrival time at the Expo line stop is equally likely to be anywhere within that 10 minute interval

(that is, you do not plan your arrival time at the station in accordance with the train schedule). Let T

denote the time to campus via metro.

a. What is the pdf of T? You may assume that when you arrive at the bus stop, it is equally like to

take anywhere between 0 and 12 minutes for the bus to arrive.

b. Suppose you ride your bike. The bike ride to campus is 35 minutes. What is the probability the

bike ride is faster than taking the metro?

Question 3. (40 points) You did not trust the distribution we gave in class to model the time to load a

web page. Specifically, you did not agree that the variance should be infinite. Therefore, you decided to

do some experimenting on your own to decide on the appropriate distribution. Your emphasis is to

determine the tail of the distribution. For that, if 􀜺 is a random variable that models the web page load

time, you decide it is sensible to assume

􀜲􁈺􀜺 > 􀝔􁈻 = 1

􀝔􀯗 for 􀝔 ≥ 1,

for some 􀝀 that you will fit to your data, so that

􀜨􁈺􀝔􁈻 = 1 − 1

􀝔􀯗 for 􀝔 ≥ 1.

Before you proceed with your data fitting, you want to understand the behavior of this family of

distributions, parameterized by 􀝀.

a. (10 points) What is the median of this distribution, as a function of 􀝀?

b. (20 points) What is the mean of this distribution, as a function of 􀝀? For what values of 􀝀 does

the mean exist?

c. (10 points) Are the median and mean equal? Interpret your answer in relation to the

distribution of times it takes a web page to load. What happens as 􀝀 becomes large (show work

–either in Excel or via algebra)?

Question 4. (20 points; 10 per part) Every week, Factory Trojan produces 1000 items on average. Some

weeks are more and some weeks are less, primarily due to yield issues.

a. Do you deem it likely that next week Factory Trojan will produce 3000 or more items? Why or

why not? (Hint: Use Markov’s inequality.)

b. Suppose you also know that the standard deviation of each week’s production is 100. Do you

deem it likely that next week’s production will either exceed 1300 or be lower than 700? Why

or why not? (Hint: Use Chebyshev’s inequality.)