Q 6.2. Let X = (X1, X2, X3)T MVN(x, Ex) where 2 -3 -() 1 fx = and Ex: = 6 -2 L - -2 - (a) Compute the moment generating function Mx (t) of X. (b) Compute E(X1 X₂). (c) Let Y₁ = 3X2 X3 + 1 Y₂ X₁ X2 X3 Y3 = X₁ + 2X₂ - 2. Compute the distribution of Y = (Y₁, Y2, Y3)T. -2 -2 2 1 1 1 -
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- Suppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N. The mgf(moment generating function) of Xiis MXi(t) = (1 − (t/β) )−αi . (a)Use the mgf of Xi to derive the mgf of ∑i=1 Xi . Determine the distribution of ∑i=1 Xi based on its mgf.Q 6.2. Let X = (X1, X2, X3)T ~ MVN(µx, Ex) where -() fx = and Ex = Y₁ Y₂ Y3 Compute the distribution of Y = (Y1, Y2, Y3)T. = = ( (a) Compute the moment generating function Mx (t) of X. (b) Compute E(X₁X₂). (c) Let = 6 -2 -2 -2 -2 2 1 1 1 3X2 X3 + 1 X₁ X₂ X3 X₁ + 2X2 - 2.Suppose that Y₁ and Y₂ are uniformly distributed over the triangle shaded in the accompanying figure. 3₂ (0, 1) (-1,0) (a) Find Cov(Y₁ Y₂). Cov(Y₁, Y₂) = (b) Are Y₁ and Y₂ independent? Yes O No (1, 0) (c) Find the coefficient of correlation for Y₁ and Y₂. P= y/₁ (d) Does your answer to part (b) lead you to doubt your answer to part (a)? Why or why not? O Even though Cov(Y₁Y₂) # 0, Y₁ and Y₂ are not necessarily dependent. Since Cov(Y₁ Y₂) # 0, we should expect Y₁ and Y₂ to be dependent. O Since Cov(Y₁, Y₂) = 0, we should expect Y₁ and Y₂ to be independent. O Even though Cov(Y₁Y₂) = 0, Y₁ and Y₂ are not necessarily independent.
- Suppose X and Y are the coordinates of a randomly sampled point in the triangle with vertices (- O), (0,1). (This means their joint PDF is a constant). Find the joint PDF of X and Y, call if f (x,y). Find fy (y)6. (Sec. 5.1) Two headlights of a car have the following joint pdf for their useful lifetimes X (the left headlight) and Y (the right headlight) ze(y+1) for r> 0.y > 0 f(x, y) 0 otherwise (a) What is the probability that the lifetime X of the left headlight exceeds 2.8? (b) Find the marginal pdfs of X and Y. Are the two lifetimes independent? Justify your answer (c) What is the probability that the lifetime of at least one headlight does not exceed 2.8?1. Let the random variable Y have pdf as f(y)A) = e"=e{(e*-1), y>0, A > 0. Show that W = {e* – 1] ~ xỉ or, equivalently, U = e' –1~}X.
- Q 6.2. Let X = (X1, X2, X3)¹ ~ MVN(µx, Ex) where -3 -(1) μx = and Σχ = (a) Compute the moment generating function Mx (t) of X. (b) Compute E(X₁X2). (c) Let Y₁ Y₂ Y3 Compute the distribution of Y= (Y₁, Y2, Y3)T. = = 6 -2 -2 = -2 2 1 3X2 X3 +1 X₁ - X₂ - X3 X₁ + 2X₂ - 2. -2 1 1Let Mx (t) = 1/(1-t), t < 1 be the moment-generating function of a random variable X. Find the moment-generating function of the random variable Y = 2X +1.Assume that X1, X2, X3 ∼ Exp (λ) are independent and evenly distributed with the distribution Exp (λ = 1). (a) Determine the distribution of Y = X1 + X2 + X3 and state its PDF f (y). (b) Determine the distribution of U = 2Y