Statistical Methods

Homework 3:

Note 1: Homeworks are due at the beginning of the "due date" class. Late hw will not be accepted, except in extraordinary circumstances.

Note 2: If you can't solve some of the problems, please come to the office hours, or email us a very specific and short question. Email will not work well or at all for involved or unclear questions.

For this homework, please complete the following problems:

Problem 1

Consider a system of 6 components pictured in the following diagram:

Note that for the system to work, all of the following have to be satisfied:

• component 1 has to work
  
• at least one of the components 2, 3, and 4 has to work
  
• at least one of the components 5 and 6 has to work

Component 1 has an exponentially distributed lifetime with a mean of 1/2 year. Components 2, 3, and 4 each have an exponentially distributed lifetime with mean lifetime of 1 year. Components 5 and 6 have exponentially distributed lifetimes with mean lifetime of 1.5 year.

a) What is the probability that the system will function uninterruptedly for at least 2 years?
b) What is the probability that the system will fail in the first 3 months of the 2nd year?

Problem 2

Let X be a normally distributed random variable with mean 3 and variance 4.

a) Let Y = 5X+2. What is the distribution of Y? What are its mean and variance?
b) Find P(Y<10). Find P(X<10).
c) What is the 99th percentile of the distribution of Y?
d) What is the 99th percentile of the distribution of X?
e) What is the distribution of W = exp(Y)? What are its mean and variance?

Problem 3

Let X1, X2, X3, X4, X5 and X6 denote the numbers of blue, brown, green, orange, red, and yellow M&M candies, respectively, in a sample of size n. According to the M&M Web site, the color proportions are p1=0.24, p2=0.13, p3=0.16, p4 = 0.20, p5 =0.13, and p6 =0.14.

a) If n = 12, what is the probability that there are exactly two M&Ms of each color?
b) For n = 20, what is the probability that there are at most five orange candies? (Hint: Treat an orange candy as a success and any other color as a failure)
c) In a sample of 20 M&Ms, what is the probability that the total number of candies that are blue, green, or orange is at least 10?

Problem 4

Please answer and provide a brief justification of your answer:

a) Can covariance between two random variables be less than -1?
b) If covariance between two random variables is negative, does their correlation have to be negative?
c) If correlation between two random variables is negative, does their covariance have to be negative?
d) Can correlation between two random variables be bigger than 1?
e) If Cov(X,Y)= 0.3, what is Cov(100X,Y)?
f) If Corr(X,Y) = 0.1, what is Cov(100X,Y)?
g) What is Cov(X,X)?
h) What is Corr(X,X)?
k) What is Cov(100X,10X)?
l) What is Corr(100X,10X)?

Problem 5

A rock specimen is randomly selected and weighed two different times. Let w denote the true weight (a number) of the rock, and let X1 and X2 be the two measured weights. Then, X1 = w + E1, and X2 = w + E2, where E1 and E2 are the two measurement errors. Suppose that E1 and E2 are independent, and distributed normally with mean 0 and variance equal to 0.1 (ie, E1, E2~ N(0,0.1)).

a) What is the mean of X1? What is the mean of X2?
b) What is Var(X1)? What is Var(X2)?
c) What is Corr(X1,X2)?