1. In a year, weather can inflict storm damage to a home. From year to year the damage is random. Let Y be the dollar value of damage in a given year. Suppose that 95% of the years Y=$1,000, and 5% of the years, Y=$15,000.
a. Compute the mean and standard deviation of the damage in any year.
b. Consider an 'insurance pool' of 100 sufficiently dispersed homes, which implies the damage to different homes can be viewed as independently distributed as random variables. If ? is the average damage to those 100 homes in a year, (i) what is the expected value of the average damage? (ii) What is the probability that ? exceeds $2000?
2. Since questions related to every day Economics involves more than one variable, let us work with 2 random variables.
Suppose the following table gives the joint PDF (probability distribution function, not Adobe document!!) of two discrete variables, x and Y.
Variable X
-2 0 2 3
3 0.27 0.08 0.16 0
Variable Y
6 0 0.04 0.10 0.35
Interpretation of the Table: If the variable X takes on a value, (-2) and the variable Y takes on a value 3, their joint probability is 0.27. In other words, the probability of X=-2 and Y=3 simultaneously is 0.27. Continuing this, the probability of X=0 and Y=3 is 0.08 and so on.
Using the information given in the table above, calculate
a) Marginal Probability Density Functions of X and Marginal Probability Density Functions of Y.
b) Conditional probability of (X=-2|Y=3) and the conditional probability of (X= 2 | Y=6).
c) Determine the expected value of X, E(X), and expected value of Y, E(Y).
d) Find the Covariance between X and Y, cov(X, Y).
e) Calculate the coefficient of correlation, r.