Let TN be given as (TN. 6) = |F(Nx) 0(x) dx, for any ø e C5 (R). Show that the distributional limit of Ty as N → ∞ is the Dirac delta distribution, that is, lim (TN, Ø) = ø(0), for any ø e Cổ°(R).

Q: 6 (b). Let 0 < f(x) < cos x and 0 < g(x) < sin x for x E I. Show that there is ro E I such that…

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Q: The graph of f(x) is given: L 1500 1000 500 -10-9-8-7-6-5-4-3-2- Ó (-∞0, ∞0) -500 -1000 -1500 Where…

A: we need to find the interval in which f(x) is increasing.

Q: Let X1, . . . , Xn be i.i.d. as the Poisson distribution Pois(λ). Find the limit φ(t) of the…

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Q: Let f(x) be uniformly continuous and non-increasing on [0, 0). Assume that Jo f(x) dx 0 for every x…

A: Given : f(x) is uniformly continuous and non increasing on [0,∞).              Also assume that…

Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

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Q: If f is a piecewise continuous and T-periodic, then show that for a e R T ra+T f (x) dx = - (2) f…

A: i used the change of variable formula to obtain solution

Q: Suppose the pdf of X is given by = {1+² 0 f(x) = (1/3+²³ × 0≤x≤2 otherwise ✪ Derive F(x). Calculate…

A: Given,f(x)={18+38x; 0≤X≤20otherwise

Q: 3. (i). Every uniformly continuous function is a Lipchitz function. (Prove or Disprove)

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Q: Let f be continuous over the interval [0,4] and let G (x) = S, f(t)dt for all x e [0,4]. a)…

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Q: 4. Determine whether or not the following functions are (a) f(x) = V¤ on (0, 0). uniformly…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Assume that the function f is such that f(3) = 1, ſ f(x) dx = 5, and f'(x) is continuous on [0, 3].…

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Q: (4) Consider I = [0, 1] with Lebesgue measure m. Assume f e L* (I) with fl =r>0. Let an GS"dm. Show…

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Q: Suppose X is continuous with f(x) = x/2 from 0 to 2. What are the values of f(1) and P(X=1)? 1/2 and…

A: Given,f(x)=x2 ; 0&lt;x&lt;2

Q: 4. Find the differentiation dy. using definition with limit of the following: y = V2x + 1.

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Q: The function f(x) = k sin 7x models a continuous probability for 0≤x≤ 17. Compute k: 7/2 Compute…

A: The function  models a continuous probability for .To find the value of ,  and .

Q: Let g be continuous on [a, b] where g(x) > 0 for all x € [a, b] and define F(t) = f g(x)dx for te…

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Q: Given e f* f(x) dx. evaluate f(x) = ²+1 HIN if 0<x< 1 if 1 ≤ x ≤ 3,

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Q: (d) f(x) = arctan x on R. (e) f(x) = 1/(1+ x²) on R.

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Q: Determine whether the following functions are uniformly continuous on the given domain via € - 8. X…

A: Any absolutely continuous function (over a compact interval) is uniformly continuous. On the other…

Q: [T(e))? Var (T) 2 where T = t(x,1.,x,) is unbiased estimator for T(0). Then the nElog f(x,e)] above…

A: Estimation and Estimator: When we are concerned in finding numerical values of unknown parameter(s),…

Q: ) For what value of a is fx² -1 x<3 S(x)= { |2ax x23 Continuous at every x?

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Q: Suppose X1,...,Xn are iid Exponential(θ = 2), such that E(Xi) = 2. Further assume that n is large…

A: Suppose X1 ,...,Xn are iid Exponential( = 2), such that E(Xi) = 2.From the central limit theorem,

Q: Suppose f is a continuous function defined on (-e, e) for some e > 0 and assume that f(0) = 0 and…

A: Since function f(x) is continuous at x=0. Also given that limx→0f(x)x=C

Q: d lim g(q)d q→7 (dg

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Q: Let X have a gamma distribution with α > 1. Show that E [1/X] = 1/[θ*(α −1)]

A: X~ Gamma(α, θ) Pdf of gamma distribution, f(x)= θαx(α-1)e-θx/Γ(α) ,  0&lt;x&lt;infinite   E(1/X)=…

Q: Let fe C[0, 1]. Show that E lim n→∞ of |f(x)|¹dx n max |f(x)\. x= [0,1]

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Q: dy -) using definition with limit of the following: dr 4. Find the differentiation (- y = /2x + 1.

A: We need to find the differentiation of y=2x+1 using the definition of limit. Now, according to the…

Q: Let Z~N(0, 1) and X ~ N(2,4). (a) Calculate E[Z]. (b) Calculate E[X4] (hint: try expressing X as a…

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Q: The pdf of a continuous RV X is given by f(x) = cos(Ax)+ 10, where x E [0, -1. Find the value 2A of…

A: Given density function of the random variable X is, f(x)=cos(Ax)+10, x∈0,π2A The value of A is…