1. Consider three fair tosses of a coin, and let X = number of heads and Y = number of changes in the sequence of toss results (e.g., HHH has no change of sequence, HTH has two changes of sequence, etc.)
a. Construct the sample space of all outcomes of this experiment and tabulate the marginal probability distributions of X and Y.
b. Tabulate the joint probability distribution of X and Y in the form of a two-way table.
c. Tabulate the conditional probability distribution of X given Y = 1.
d. Find the values of E(X), Var(Y), E(Y), Var(Y), and Cov(X,Y).
e. Find the values of E(X|Y=1) and Var(X|Y=1).
f. Find the value of Var(2X-3Y).
g. Are X and Y uncorrelated? Are X and Y independent? Explain your answers.
2. Mr. Slick makes the following offer to you. You can toss four coins (furnished by him), and he will pay you an amount equal to the square of the number of heads showing (e.g., if you get three heads, he will pay you $9). In order to play this game you must pay $4 for each four-coin toss. On the naive assumption that the coins are fair, what is your expected gain?
3. Let X be a normally distributed variable with mean µ and variance σ2. A random sample of three observations was obtained from this population. Consider the following estimators of µ.
µ^1 = (X1 + X2 + X3/3) and µ^2 = X1/6 + X2/3 + X3/2
a. Derive the expected value and variance of each estimator.
b. Compare the properties of these estimators in terms of unbiasedness, efficiency, and consistency.
4. You are told that X ~ N(8,36). Based on a sample of 25 observations, it was found that X- = 5.7
a. What is the sampling distribution of X-?
b. What is the probability of obtaining an X- = 7.5 or less?
c. Construct a 95% confidence interval for the population mean.