Show that ø(n²) = nø(n) for every n E Z*.

Q: 12. Define def M e-$dWg• s=0 For each t > 0, compute (a) E[M¡]. (b) E[M?]

A: Given  Mt=∫s=0te-s dWs      , where W is a Brownian motion.

Q: 1. Show (in terms of e- 6) that a function f : R* → R defined by f(x, y, 2) = (2x + 3y + 42) is…

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Q: 2. Let {N(t),t 2 0} be a renewal process and (X,,X,, X,.-be the corresponding interarrival times.…

A: A renewal process is a stochastic model that occurs at random and is temporal based. The process…

Q: Show that for some number a > 1, we have 1- <= In(a) <= a – 1. Hint: In(a) = S"dx, please check the…

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Q: Let X1 ∼ Gamma(k1, λ) and X2 ∼ Gamma(k2, λ) independently. Show that X1/(X1 + X2) ∼ Beta(k1, k2).…

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Q: 4. The total number of the accessible microscopic states of the Boltzmann gas, with energy E and…

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Q: dn-1 for all n E Z with n > 2. Prove that for a 2. Let d = 2, and d, neN, d, =. %3D %3D %3D

A: Mathematical Induction is a special way of proving things. It has only 3 steps: Step 1: Show it is…

Q: 2. If {X(t); te T} be a time series then for any t1, t2 define the autocovariance function (t1,t2) =…

A: Given Information:   Xt:t∈T be a time series for any t1 and t2.

Q: If f(z, y) – e . In(9y) , then fz is given by: O - 9e In(9y) o - 27 e In(9y) O 9 e In(9y) O 27 e…

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Q: (a) Let X have the following pdf : S2(x - 1) 1<z<2 S(2) = elsewhere. Compute E(X'), E(2X-1)

A: As per our guidelines we are suppose to answer first question.

Q: Let Zn be the size of the nth generation in a GW process with Zo = 1, E(Z¡) = µ. Show that E(Z„Zm) =…

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Q: LetX ∼ P oisson(λ). Prove that E[X(X −1)] = λ 2 and use this result to show that var(X) = λ.

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Q: 7) Use Theorem 7.1.1 (Table) to find L{f(1}} for the following: S)-(e*-e*)}° a.

A: The given function is f(t)=(e-3t-e5t)2. Expand the function implies, f(t)=e-6t+e10t-2e2t The Laplace…

Q: .3 T Evaluate In In e²+j²4

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Q: Find the constant term in the expansion

A: We have to solve the problem

Q: Prove that for the gamma function [(x) = ftx-¹e-tdt that for n a positive integer then I(n) = (n-1)!

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Q: 1E(X; – Xn)². Prove 1. Let X1,..., Xm,... - iid N(u, T) and consider the MLE în that î, is a…

A: Given X1, X2, … Xn iid N(µ, τ) Given MLE

Q: 6. Let {fn} be a sequence of measurable functions on E = (0, 1] satisfying %3D Prove that the…

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Q: 1. Let g(x, y) = ev* /x. Evaluate gry and gyæ.

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Q: Suppose that T(X) is sufficient for 0 and that ^e(X) is an MLE of 0. Show that ^0 depends on X…

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Q: Show (in terms of e - 6) that a function f : [2,7] R be defined by f(z) = V² +1 is uniformly…

A: In this question, we need to show the given function f(x) is uniformly continuous.…

Q: Show that the series n=0 xn cos(nx) x² +n³ + 1 defines a differentiable function g(x) for x in [-1,…

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Q: If Σ α An diverges then E ca can diverges only if c > 0 only if c < 0 only if c+ 0 for any real…

A: Given, ∑an diverges then find for which values of c, ∑can diverges.

Q: Suppose that 0< fn < In on X for all n e N. If gn 30 on X, then fn 30 on X.

A: Solution:-

Q: If Σ An diverges then > can diverges O only if c>0 O only if c< 0 O only if c+0 O for any real value…

A: Given, the series ∑an diverges, then we have to find the values of c for which the series ∑can…

Q: Show that the random process X(t) = A cos (@n t+ 0) is wide-sense stationary if it is assumed that A…

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Q: Find y' for each of the following. e2x+3y = x2 - In %3D

A: Answer and explanation is given below

Q: 2. Z=e Iny

A: Given that, z = ex ln y x = ln (u cos v) y = u sin v We have to fin dz/du and dz/dv. Then we get,…

Q: Find the limit of the sequence bn = (cos n)/(n2 + 1).

A: Given sequence is bn=cosnn2+1.

Q: Suppose f"(x) = N (h) + a,h + azh2 + azh + -. The value of N½(h) using Richardson's extrapolation…

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Q: lim, »∞(e² + x)'/z =? O a. 1 O b. Limit does not exist. O cO O d. e O e. 00

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Q: 11. Prove the absorption law x +xy=x using the other laws in Table 5.

A: To prove: x+xy=x We use the following identities (1) x=x·1 (2) 1+x=1 (3) xy+z=xy+xz    ( This is…

Q: Consider the following recurrence T : N → N: T(0) = 0 T(n) = T(L) +T(L4J)+n_for n > 1 What can you…

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Q: 2.44 Suppose f is uniformly continuous on D and suppose E c D. Prove: f is uniformly continuous on E…

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Q: by

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Q: 4. Exercise §3.4 #47. Let n €Z with n> 0. Prove that E - a(n)

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Q: and further we require that u(r, z) is bounded as z - +oo. Find an expression for the steady state…

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Q: Find lower bound for f(n) = 5n².

A: To find the lower bound for f(n) = 5n2

Q: Suppose that an initial deposit of $100 is made, and a deposit of X is made at t = 2. Find X if…

A: Given that : Principal value = $100  when t = 2. Amount = $350  when t = 4 To find X = ? To use the…

Q: For the following PMFS X and Y for k e {0,1,2,3, ..] Px (k) = { %3D otherwise for k e {0,1,2,3, ...}…

A: From the given information, PXk=e-λλkk!, k=0,1,2,3,....PYk=e-λλkk!, k=0,1,2,3,.... If λ=1, a) The…

Q: Find fCx) if flo)=0 and the tanGat (k,F CX) has ine af Slore x(x+2) a

A: Given for the function F(x), F(0)=0 and tangent line to F(x) at x,F(x) has slope xx2+23.

Q: Prove that by Gamma Function eit + e' e-st dt 2 s2 + 1

A: We will find out the required result using Gramma function.

Q: Does E 8. k=1 In(ek41) converge or diverge?

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Q: Let f(x) = En-0 n, > 1 Compute E2(f(k) – 1) using doubly indexed sequences.

A: Consider the provided question, In your problem n=0 to ∞ is not define at n=0fx=∑n=1∞1nx,     x&gt;1…

Q: 5. Show that |Logz|<|In(z)|+ t, for all z # 0.

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Q: Prove that e“ p.w. on R. n! n=0 M8

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Q: Prove that (A')' = A for each A E Mmxn(F).

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Q: Find the PSD of a random process X(t) if E[X(t)] = 1 and R(T) = 1 + e-1T| x,

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