stant) and find the limit of My (t) as n →∞. What is the limit of the corresponding distributions? stify your answers.
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- Take the model y = y x'1β1+x'2β2+ϵ with E(xe)=0 Suppose B1 is estimated by regressing Y on X1 only. Find the probability limit of this estimator.Problem 4. Let Φ be the standard distribution function. Show that Φ(B(t)) / √ T − t is a martingale for 0 ≤ t < T using Itô-formula. please use definition and reasons while solving the problem.Problem 11 The volume of the solid obtained by rotating the region enclosed by y=(x^2), y=6x, about the line x=0 can be computed using the method of disks or washers via an integral V=∫[upper limit=b, lower limit=a]________dy with limits of integration a=_______ and b=________
- Consider a birth and death process, X = {X(t)} : t ≥ 0 with the 4 states, S = {0, 1, 2, 3} and instant rates per hour are specified implicitly as follows: 1 /λk = 10 minutes for (0 ≤ k ≤ 2) and 1/ µk = 30 minutes for (1 ≤ k ≤ 3) Assume that λk = 0 for k ≥ 3 and µk = 0 for k > 3. a) Derive the limiting distribution, lim t→ ∞ P [X(t) = k] for all k b) Evaluate expectations in minutes for departure times, E [S1 |X(0) = 0] and E [S3 |X(0) = 3]Problem 21 Given the equation xy=12, set up an integral to find the length of path from x=a to x=b and enter the integrand below. L= ∫(upper limit=b, lower limit=a)__________________________ dxTheorem 6.4 states that the moment-generating function of the gamma distribution is given by Mx(t) = (1-βt)^(-α).
- If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?Problem 9 The volume of the solid obtained by rotating the region bounded by x=(9y^2), y=1, x=0 about the y-axis can be computed using the method of washers or disks via an integral V=∫________________dy (with lower limit of a and upper limit of b) with limits of integration a=_________ and b=_________ The volume of this solid can also be computed using cylindrical shells via an integral V=∫_______________dx (with lower limt of alpha and upper limit of beta) with limits of integration alpha=_________ and beta=__________ In either case, the volume is V=______________ cubic unitsIf X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).
- Problem 12 The volume of the solid obtained by rotating the region bounded by y=(x^2), y=5x about the line x=-2 can be computed using the method of washers or disks via an integral V=∫________________dy (with lower limit of a and upper limit of b) with limits of integration a=_________ and b=_________ The volume of this solid can also be computed using cylindrical shells via an integral V=∫_______________dx (with lower limt of alpha and upper limit of beta) with limits of integration alpha=_________ and beta=__________ In either case, the volume is V=______________ cubic unitsProblem 10 The volume of the solid obtained by rotating the region enclosed by y=(x^2), y=2x, about the line y=0 can be computed using the method of disks or washers via an integral V=∫[upper limit=b, lower=a]____________dx with limits of integration a=________ and b=________Problem 12 The volume of the solid obtained by rotating the region enclosed by x=3y, y^3= x (with y>/=0) about the y-axis can be computed using the method of disks or washers via an integral V=∫[upper limit=b, lower limit=a]_______dy with limits of integration a=______ and b=_______