The probability distribution of the number of students absent on Mondays, is as follows: X 0 1 2 3 4 5 6 7 f(x) 0.02 0.03 0.26 0.34 0.22 0.08 0.04 0.01 (a) What is the probability that more than 3 students are absent. (b) Compute the expected value of the random variable X . Interpret this expected value. (c) Compute the variance and standard deviation of the random variable X . (d) Compute the expected value and variance of Y=7X+3 . (e) Compute the covariance Cov(X, Y) (f) What is the value of the ratio Cov(X, Y)/Var(X) ? (g) What is the value of Cov(X, Y)/Var(X) for Y=β0+β1X ?

a) Add up all the probabilities where :

b) The expected value is

Since X is the number of absent students on Monday, the expectation E[X] is the number of students you can expect to be absent on average on any given Monday. According to the distribution, you can expect around 3 students to be consistently absent.

c) The variance is

where

So the variance is

The standard deviation is the square root of the variance:

d) Since is a linear combination of , computing the expectation and variance of Y is easy:

e) The covariance of X and Y is

We have

so

Then the covariance is

f) Dividing the covariance by the variance of X gives