1. Let K be the index of the first successful trial that is immediately followed by...

1. Let K be the index of the first successful trial that is immediately followed by a failure. In symbols, (3.53) K = inf{n ∈ Z>0 : Xn = 1, Xn+1 = 0}. Find the probability mass function of K. Check that your answer is a legitimate probability mass function. Hint. Decompose the event {K = m} into disjoint components expressed in terms of the trial outcome variables {Xi}. Note that a success before m cannot be immediately followed by a failure.

2. Let N be the index of the first success, as defined in (3.18). Let K be the index of the first success that is immediately followed by a failure, as defined in (3.53). (a) Find the probability P(XN+1 = 0, XN+2 = 1, XN+3 = 0). (b) Find the probability P(XK+1 = 0, XK+2 = 1, XK+3 = 0). The task is to find the probability that the three trials immediately following the random index K yield a failure, a success, and a failure, in that order. Explain why your answer makes intuitive sense.

3. Assume that 0 < p < 1. (a) Let Sn ∼ Bin(n, p) count the number of successes in the first n trials. Fix a positive integer k. Show that lim n→∞ P(Sn ≤ k) = 0. (b) Show that in infinitely many trials there are infinitely many successes with probability one.

Expert Solution

Derivation of the PMF of K

Derivation of the PMF of K (Continued) + Proof that it is a valid PMF
The PMF of K was derived as :

Proof that it is a valid PMF (Continued) + Solution to 2.(a)

Solution to 2.(a) (Continued) + Solution to 2.(b) + Solution to 3. (a)

Solution to 3.(a) (Continued) using Hoeffding's inequality + Solution to 3.(b)

Please upvote and provide feedback if this answer helped you. This would help me improve and better my solutions.
I will be happy to answer your doubts, if any in the comment section below. Thanks! :)

orchestra answered 2 years ago

Let S_k(n) = 1^k + 2^k + ··· + n^k for n, k ≥ 1. Then,...

Let S_k(n) = 1^k + 2^k + ··· + n^k for n, k ≥ 1. Then, S_4(n) is given by S_4(n)= n(n+1)(2n+1)(3n^2 +3n−1)/ 30 Prove by mathematical induction.

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let W be a discrete random variable and Pr(W = k) = 1/6, k = 1,...

Let W be a discrete random variable and Pr(W = k) = 1/6, k = 1, 2 ,....., 6. Define X = { W, if W <= 3; 1, if W >= 4; } and Y = { 3, if W <= 3; 7 -W, if W >= 4; } (a) Find the joint probability mass function of (X, Y ) and compute Pr(X +Y = 4). (b) Find the correlation Cor(X, Y ). Are X and Y independent? Explain.

Let S = {2 k : k ∈ Z}. Let R be a relation defined on...

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class

Let k be an integer satisfying k ≥ 2. Let G be a connected graph with...

Let k be an integer satisfying k ≥ 2. Let G be a connected graph with no cycles and k vertices. Prove that G has at least 2 vertices of degree equal to 1.

Doug just received a settlement from a lawsuit that pays him $125,000 immediately, followed by amounts...

Doug just received a settlement from a lawsuit that pays him $125,000 immediately, followed by amounts of $125,000 at the end of each year over the next 10 years with one exception – lawyer fees reduce the amount in Year 3 to $75,000. However, the lawsuit allows Doug to receive the equivalent present value today. Using an interest rate of 6.5%, which of the following comes closest to the equivalent present value? a. $1,200,000 b. $ 857,145 c. $ 898,604...

Theorem: Let K/F be a field extension and let a ∈ K be algebraic over F....

Theorem: Let K/F be a field extension and let a ∈ K be algebraic over F. If deg(mF,a(x)) = n, then 1. F[a] = F(a). 2. [F(a) : F] = n, and 3. {1, a, a2 , ..., an−1} is a basis for F(a).

QUESTION: In the case of liquidation, common stockholders are paid first, followed by preferred stockholders, followed...

QUESTION: In the case of liquidation, common stockholders are paid first, followed by preferred stockholders, followed by bondholders. ANSWER OPTIONS: True False You need to specifically state IN THE SUBJECT LINE if the answer is TRUE or FALSE. EXAMPLES OF INADEQUATE RESPONSES: “I think the answer is False.” OR “The correct answer is “C.” Postings must be no less than 200 words in length to be considered. Any posting less than 200 words in length will not be reviewed.

Let K = { s+t * 2^(1/2), such that s, t are Rational}. Show that K...

Let K = { s+t * 2^(1/2), such that s, t are Rational}. Show that K is a Field

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn...

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn where k ≤ n. (a) For σ ∈ Sn, prove that στσ-1 = (σ(1), σ(2), . . . , σ(k)). (b) Let ρ be any cycle of length k in Sn. Prove that there exists an element σ ∈ Sn so that στσ-1 = ρ.

Question