Consider three fair tosses of a coin and let X number of hea

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.

Solution

a.

Sample Space = {HHH,HTH,HHT,THH,HTT,THT,TTH,TTT}

There are a total of 8 items in sample space.

Marginal Distribution of X

Marginal Distribution of Y

b.

Joint distribution Table of X and Y

c.

d.

E(X) = 0*(1/8) + 1*(3/8)+2*(3/8)+3*(1/8) = 1.5

Var(X) = (0-1.5)²(1/8) +(1-1.5)²(3/8) +(2-1.5)²(3/8) +(3-1.5)²(1/8) = 0.75

E(Y) = 0*(2/8) +1*(4/8) +2*(2/8) = 1

Var(Y) = (0-1)²(2/8)+(1-1)²(4/8)+(2-1)²(2/8) = 0.5

Cov(X,Y) = (0-1.5)(0-1)(1/8)+(3-1.5)(0-1)(1/8) + (1-1.5)(1-1)(2/8)+(2-1.5)(1-1)(2/8) + (1-1.5)(2-1)(1/8)+ (2-1.5)(2-1)(1/8) = 0

Please ask rest questions seperately.