Mr. and Mrs. Brown decide to continue having children until they either have their first girl or until they have three children. Assume that each child is equally likely to be a boy or a girl, independent of all other children, and that there are no multiple births. Let G denote the numbers of girls that the Browns have. Let C be the total number of children they have.
(a) Determine the sample space, along with the probability of each sample point.
(b) Compute the joint distribution of G and C. Fill in the table below.
C=1
C=2
C=3
G=0
G=1
(c) Use the joint distribution to compute the marginal distributions of G and C and confirm that the values are as you’d expect. Fill in the tables below.
P(G=0)
P(G=1)
P(C=1)
P(C=2)
P(C=3)
(d) Are G and C independent?
(e) What is the expected number of girls the Browns will have? What is the expected number of children that the Browns will have?
CS 70, Fall 2018, HW 10 1
2 Will I Get My Package?
A delivery guy in some company is out delivering n packages to n customers, where n∈N, n 1. Not only does he hand a random package to each customer, he opens the package before delivering it with probability 1/2. Let X be the number of customers who receive their own packages unopened.
(a) Compute the expectation E(X).
(b) Compute the variance var(X).
3 Double-Check Your Intuition Again
(a) You roll a fair six-sided die and record the result X. You roll the die again and record the result Y.
(i) What is cov(X+Y,X−Y)?
(ii) Prove that X+Y and X−Y are not independent.
For each of the problems below, if you think the answer is "yes" then provide a proof. If you think the answer is "no", then provide a counterexample.
(b) If X is a random variable and var(X)= 0, then must X be a constant?
(c) If X is a random variable and c is a constant, then is var(cX)=cvar(X)?
(d) If A and B are random variables with nonzero standard deviations and Corr(A,B)= 0, then are A and B independent?
(e) If X and Y are not necessarily independent random variables, but Corr(X,Y)= 0, and X and Y have nonzero standard deviations, then is var(X+Y)= var(X)+var(Y)?
(f) If X and Y are random variables then is E(max(X,Y)min(X,Y))=E(XY)?
(g) If X and Y are independent random variables with nonzero standard deviations, then is
Corr(max(X,Y),min(X,Y))= Corr(X,Y)?
