qu2 .01 ai anoitoni lo oonoupse a od (A oels bnnI.031 moitonl s of 1. Let m*(A) = 0. Prove that for every B, m* (BUA) = m*(B). %3D
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A: (note :- here we consider x1 =a, x2 = b, x3 =c and x4 = d)
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Q: (b) Justify why âx och T are linear transformations.
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Q: 9. Consider an inhomogeneous differential equation f"(x) = e". || a) 'Guess' a solution of the inhom...
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Q: -4 9 4. Figure out whether the matrix B = is diagonalizable or not (i.e. whether R? -4 8 admits a ba...
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- 2. Give the condition which ensure that |ez| < 1 where z in C.Suppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50
- dik koordinatylarda x,y öyleki x kare artı y kare küçüktür beş denkleminin grafiğini çizinizConsider the seriesxt = sin(2πU t),t = 1, 2, . . ., where U has a uniform distribution on the interval (0, 1). Prove xt is weakly stationaryLet U₁,..., Un be i.i.d. random variables uniformly distributed in [0, 1] and let Mn = max Ui . 1<i<n Find the cdf of Mn, which we denote by G (t), for t = [0, 1]. For t = [0, 1], G (t) = Now, let Fn (t) denote the cdf of n (1 - Mn); for t > 0, compute lim F₁ (t) =
- Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentConsider a random process X(t) defined by X(t) = U cos t + (V + 1) sin t, −∞ < t < ∞where U and V are independent random variables for which E(U) = E(V) = 0 E(U2) = E(V2) = 1(a) Find the autocovariance function KX(t, s) of X(t).(b) Is X(t) WSS?Show that if φ is continuously differentiable in a given region V and on itsboundary S, then∫S φ dS =∫V ∇φ dV
- Suppose P(X = 2) = 1/2, P(X = 5) = 1/3 and P(X = 7) = 1/6(a) Compute the probability generating function rX(t)(b) Verify that r'X(0) = P(X = 1) and rX”(0) = 2P(X = 2)(c) Compute the moment generating function of X, mX(s)(d) Verify that m'X(0) = E(X) and mX”(0) = E(X2)If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)Suppose X is a continuous random variable with density f(x) = x/2 , 0 <= x <=2 f(x) = 0 , elsewhere Write an integral expression for the moment generating function M(t).