1. n~m in Z if n-m+nm ≥ 0.

Q: n(n+1)(2n+1) 2.Prove k2 = for n > 1. 6. k=1

A:

Q: 11. Consider the expansion of 3x+ where p> 0. The coefficient of the term in x is equal to the…

A:

Q: 1. Determine L{t*}. n! Ans. s*+1

A: We find the L{tn} using definition of Laplace Transform. We know that,

Q: Suppose U ~ Binomial(n, p) and n > 2. What is P(U > 1)? Select one: 1- (1- p)"-[1 + (n – 1)p) O b.…

A: Given that ; U follows binomial distribution.  We want to find P( U> 1) =?  So by using binomial…

Q: Based on Theorem 4.4.3 P.169, when n > 2, which of the following is biggest? 1. n! 2. n^n 3. (n!)^2…

A: We have to find biggest number among of them

Q: 8. Let {X, r≥ 1} be independent with the distribution of the preceding Un = n¹"=1 Xr, and Tn = n¹Un.…

A:

Q: 7. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2,…

A: X1,X2,...,Xn is a random sample from a Bernoulli distribution with parameter pYn=1n∑Xi,  is the…

Q: 18. Let T) Σ-1() k=0 (a) Use Pascal's Triangle to compute T(n) for n = 1, 2, 3, 4. (b) Prove that…

A: We have

Q: Show in detail that E [(X-µ)^n] = 0 for ∀n , where X~N (µ, σ²) and n≥3.

A: It is an important part of statistics. It is widely used.

Q: 14. let X be the number of times a 3 appears in 600 throws of a die. (a) Find the pmf. (b) Find E…

A: Dear students as per guidelines we answer the first question only when multiple different question…

Q: 6. An unfair coin is flipped repeatedly, where P(H) = p = 1- q. Let X be the number of flips until…

A:

Q: Q₁: For (xy) e ZxZ, we say that x=ymodn if n/x-y. as Show that {(x, y) € Zx Z = x = y modo} = {(xy)…

A: (a) Let,           (a, b) ∈ {(x, y) ∈ Z×Z : x ≡ y mod 6}         => a ≡ b mod 6         => 6 |…

Q: 4,452,8,... 256 given seguence Tn=arn-l 256=4x J2 solue for n? %3D

A:

Q: 10. What is the modulus of z if z = 120_ 19 +1? (a) √2 (b) √3 (c) √4 (d) √5

A:

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: 69. Use the principle of mathematical induction to prove that dk : (In x) = (–1)k-1 «– D! for k > 1.…

A: Use the principal  induction to prove thatdkdxk(ln x)=(-1)k-1(k-1)!xk , for k≥1.

Q: 5. M = 3₁ 0² = 9₁ P/²6 < X < 12) a. Determine Plμ -ko < X < µ + ko ) 2 1 - 1 / / / k²

A: P(-6&lt;X&lt;12)=P(-6-33&lt;X-μσ&lt;12-33)=P(-3&lt;Z&lt;3)=P(-3&lt;Z&lt;0)+P(0&lt;Z&lt;3)=0.4987+0.4…

Q: 3. Prove that - e 2bt - e n! n=0 Hint: See section 3.2.

A: Given: 121-e-bt∑n=0∞nn-1n!sebtn→(1)

Q: 13. Mark the statement True or False. Justify your answer. (a) If {sn} diverges, then {Sn + Sn+1}…

A: Assume sn=n and tn=n be the divergent sequences.

Q: 19. Prove that 2n> n(n−1)(n−2) for all n Є N. 6

A: To prove: 2n&gt;6n(n−1)(n−2)​  for all n in the set of natural numbers (N), we can use mathematical…

Q: The largest inte which th (In(e- e)) y + y = 1, y(5) = 0 %3D x -6 has a unique solution is (In (1 +…

A:

Q: Show that E [(X-µ)^n]=0 for ∀n, with X~N(µ, σ²) and n≥3 being odd numbers.

A: We need to show; E[(X-μ)n]=0 for ∀n and n≥3 being odd numberswhere;X~Normal(μ,σ2) That is, we need…

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: 1. E=4 diverges, En=4 n-3 diverges. SO n

A: Concept:

Q: Prove that Z(Sn) = {e} for n ≥ 3.

A:

Q: 16. prove that Edn 2(d) = 2u(n). 17. prove that Edn H(n/d)T(d) = 1 and Edn H(n/d)o(d) = n for all n,…

A:

Q: 1. Customers arrive randomly at a bank teller's window. Suppose that exactly one customer arrived…

A: From the provided information,X~Unif (0, 10)Here X equal the time within the 10 minutes that the…

Q: 13. Prove E-1(4i – 3) = n(2n – 1) for natural numbers n > 1 %3D

A:

Q: Compute the following: remember to show your working. (a) N An if An = (1 – ,n] + (e.g. x E A3 →} <…

A: We will partition An and then find the intersection of all An as n tends to infinity.

Q: Consider the sum S(n) where i=n S(n) = Σ i=1 2°5 Find a constant real number C> 0 (i.e. C can't…

A: It is given that Sn=∑i=1ni5 such that Sn≥Cn6 for all n&gt;0.

Q: Compute ΣΣ k=0 n=0 (-1)n+knk 5nk!

A:

Q: Suppose that an urn initially contains r red and b blue balls. At each stage a ball is randomly…

A: Given, number of red balls initially in urn = r number of blue balls initially in urn = b At the…

Q: distributed random variables with Var(X1) < ∞. Show that n 1 E;X; →p EX1. п(п + 1) j=1 Note. A…

A:

Q: 1. A fair coin is tossed n times, showing heads Hn times and tails Tn times. Let Sn = Hn — Tn. Show…

A:

Q: Under normality, prove the followings (a) B~ N(B,0²(X'X)-)

A: It is known that β^=X'X-1X'Y. Here, Y follows a multivariate normal distribution. The regression…

Q: Compute the following: remember to show your working. 00 (a) N An if A, = (1– 4, n] (e.g. x € A3 +…

A: Union and intersection of sets