[SOLVED] Probability & statistics for eecs homework 14

1. A DNA sequence can be represented as a sequence of letters, where the “alphabet” has
4 letters: A,C,T,G. Suppose such a sequence is generated randomly, where the letters
are independent and the probabilities of A,C,T,G are p1, p2, p3, p4, respectively.(a) In a DNA sequence of length 115, what is the expected number of occurrences
of the expression “CATCAT” (in terms of the pj )? (Note that, for example, the
expression “CATCATCAT” counts as 2 occurrences.)(b) For this part, assume that the pj are unknown. Suppose we treat p2 as a Unif(0, 1)
r.v. before observing any data, and that then the first 3 letters observed are
“CAT”. Given this information, what is the probability that the next letter is C?2. Let X1, . . . , Xn be i.i.d. r.v.s with mean µ and variance σ2, and n ≥ 2. A bootstrap
sample of X1, . . . , Xn is a sample of n r.v.s X∗
1 , . . . , X∗
n formed from the Xj , ∀j ∈
{1, . . . , n} by sampling with replacement with equal probabilities. Let X¯ ∗ denote the
sample mean of the bootstrap sample:
X¯ ∗ = 1
n
(X∗
1 + · · · + X∗
n).
(a) Calculate E(X∗
j ) and Var(X∗
j ) for each j ∈ {1, . . . , n}.(b) Calculate E(X¯ ∗|X1, . . . , Xn) and Var(X¯ ∗|X1, . . . , Xn).
Hint: Conditional on X1, . . . , Xn, the X∗
j , ∀j ∈ {1, . . . , n} are independent, with a
PMF that puts probability 1/n at each of the points X1, . . . , Xn. As a check, your
answers should be random variables that are functions of X1, . . . , Xn.(c) Calculate E(X¯ ∗) and Var(X¯ ∗).
(d) Explain intuitively why Var(X¯) < Var(X¯ ∗).3. A coin with probability p of Heads is flipped repeatedly. For (a) and (b), suppose that
p is a known constant, with 0 < p < 1.
(a) What is the expected number of flips until the pattern HT is observed? What
about the pattern HH? Solve the problems using conditional expectation.(b) Now suppose that p is unknown, and that we use a Beta(a, b) prior to reflect our
uncertainty about p (where a and b are known constants and are greater than 2).
In terms of a and b, find the corresponding answers to (a) and (b) in this setting.4. A fair 6-sided die is rolled repeatedly.
(a) Find the expected number of rolls needed to get a 1 followed right away by a 2.
(b) Find the expected number of rolls needed to get two consecutive 1’s.(c) Let an be the expected number of rolls needed to get the same value n times in a
row (i.e., to obtain a streak of n consecutive j’s for some not-specified-in-advance
value of j). Find a recursive formula for an+1 in terms of an.
(d) Find a simple, explicit formula for an for all n ≥ 1. What is a7 (numerically)?5. Let X be the height of a randomly chosen adult man, and Y be his father’s height,
where X and Y have been standardized to have mean 0 and standard deviation 1.
Suppose that (X, Y ) is Bivariate Normal, with X, Y ∼ N (0, 1) and Corr(X, Y ) = ρ.(a) Let y = ax + b be the equation of the best line for predicting Y from X (in the
sense of minimizing the mean squared error), e.g., if we were to observe X = 1.3
then we would predict that Y is 1.3a + b. Now suppose that we want to use Y to
predict X, rather than using X to predict Y . Give and explain an intuitive guess
for what the slope is of the best line for predicting X from Y .(b) Find a constant c (in terms of ρ) and an r.v. V such that Y = cX + V , with V
independent of X.
(c) Find a constant d (in terms of ρ) and an r.v. W such that X = dY + W, with W
independent of Y .
(d) Find E(Y |X) and E(X|Y ).
(e) Reconcile (a) and (d), giving a clear and correct intuitive explanation.