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- Suppose X, Y joint Gaussian with s.d.'s 2 and 3 respectively and covariance 4. Find the joint pdf and the two marginal pdf's. What if the covariance were 8?Thank youPlease answer all part : Let rt be a log return. Suppose that r0, r1, . . . are i.i.d. N(0, 0.01^2). (a) What is the distribution of rt(8) = rt + rt−1 + rt−2 +...+ rt−7? (b) What is the covariance between r7(3) and r9(3)? (c) What is the conditional distribution r17(3) given that r16 =0.004 (d) What is the probability that the gross return over the first 10 times periods is at least 1.05?
- 2.2.5. Let X1 and X2 be continuous random variables with the joint probability density function fx1,x2(¤1,x2), Y2 = X2. -00 < xi < o, i = 1,2. Let Y1 = X1 + X2 and (a) Find the joint pdf fy,Y2· (b) Show that fy, (yı) = | fx,,X2 (y1 – Y2; Y2) dy2, (2.2.5) which is sometimes called the convolution formula.5. Two ballpoint pens are selected at random from a box that contains 3 blue pens, 2 red pens and 3 green pens. If X is the number of the blue pens and Y is the number of the red pens selected, find the joint pdf f(x, y)(47) Let X z b(8,-) find E(5+6x) and distribution function.
- 2. Let X be a continuous random variable with probability density function (pdf) 3x�� 0Suppose 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.3. Let X and Y be two continuous random variables with joint PDF of + 03. Let X and Y be continuous random variables with joint PDF (3x 0s ysxs1 f(x, y) = {* otherwise Determine the correlation of variables X and Y.3. Suppose that X is a continuous random variable with pdf 3x², 0Answer all three questions in this section 2. Let V and W be two random variables with joint probability density function given by fvw (v,w) = aw exp(-dv +(8-A)w²), w>0, v> w², where A, & are positive constants. (a) Find the value of a. (b) Consider the transformation X = WP, Y=V_W². Find the joint density of X and Y. Hence show that X and Y are independent exponential random variables. (c) State the distribution of AX + SY.Recommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning