Your Turn 3b. a. Show that variance( - (T ) - (x))2 b. Show for the...
Your | a. Show that variance(x)=(x2)-((x))2. Turnb. Show for the uniform distribution (Equation 3.5) that variance(x) a/12. 3B
just part a plz thank u! Page 4 Marks Suppose Z(t) Y., where X(t) is the Poisson process with rate θ If μ = E[h] and σ2-Yar determine the mean and variance of Z(t) a. pil are the common mean and variance for y,y , then b. fVis Uniform distribution on interval (0, 1], then determine the mean and variance of XCV) 2 Page 4 Marks Suppose Z(t) Y., where X(t) is the Poisson process with rate θ If μ...
U means Uniform distribution 2. Let X be a r.v. distributed as U(α, β). Show that its ch. f. and m.g.f.x and Mr, respectively, are given by and IM x it(β-a) , t(B-a) ii) By differentiating (ax, show that E(X)-(α + β) / 2 and T 2 (X)-(α-β)2 / 12.
C2.3 Let X and Y be random variables with finite variance, so that EX2o0 (i) Show that E(X) - (EX) E(X - EX)2, and hence that the variance of (ii) By considering (|XI Y)2, or otherwise, show that XY has finite expecta- (iii) Let q(t) = E(X + tY)2. Show that q(t)2 0, and by considering the roots of and EY2 < oo. X is always non-negative. tion the equation q(t) 0, deduce that
(b) Determine i(t) for t 20 for the circuit in Figure 3b when i(0) =-4 A and Vs(t) the voltage shown in Figure 3c. i(t) i(t) 4V 1()(X 2 3 (s) -1V Figure 3b Figure 3c
If X(t), t>=0 is a Brownian motion process with drift mu and variance sigma squared for which X(0)=0, show that -X(t), t>=0 is a Brownian Motion process with drift negative mu and variance sigma squared.
2. Let X ~ Exp(3), that is, fx (x) = t-, where x 〉 0 (and 0 otherwise). (a) Use the method of distribution functions to find the distribution (that is, pdf and (b) Use the method of distribution functions to show that Z e (c) Repeat the verification in part (b), but now using the method of transformations. domain) for Y X -3. has a uniform distri- bution. Include a description of the domain associated with the distribution
2. Let X ~ Exp(3), that is, fx (x) = t-, where x 〉 0 (and 0 otherwise). (a) Use the method of distribution functions to find the distribution (that is, pdf and (b) Use the method of distribution functions to show that Z e (c) Repeat the verification in part (b), but now using the method of transformations. domain) for Y X -3. has a uniform distri- bution. Include a description of the domain associated with the distribution
Problem 2. Solve the following ID wave equation: Ott(x,t) = 0xx(x, t) with the boundary conditions 0(0,t) = 0x(1,t) = 0, where 0(x, t) refers to the twist angle of a uniform rod of unit length. Problem 3. Show that the solution of the partial differential equation (Laplace equation),
Kindly provide calculation and show each step. Thanks (a) Given the signal x(t) in Figure 2, sketch the signal for: xit) 12 10 - 2- 0.2 0.4 0.6 2 14 1.6 1.8 2 0.8 Figure 2 i. x(t+2) [4 Marks] ii. x(t/3) [4 Marks] iii. x(2t) [4 Marks] iv. -X(t) [4 Marks] (b) Given the signal x(t) in Figure 3, sketch the signal for x-(2t-2): xit 25 2 1.5 1 0.5 -0.5 0.5 1 1.5 2 2.5 3.5 Figure 3...