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- 7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).Determine the value of c that makes the function f(x, y) = c(x + y) a joint probability mass function over the nine points with x = 1,2,3 and y = 1,2,3. Give exact answer in form of fraction.The joint density function of two continuous random variables X and Y is fxy(x, y) = (2x+y) if 2The probability density function (p.d.f.) of a continuous random variable X is defined to be: kx + for 0 < x < 1 f(x) = |0 otherwise, for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals. Part a) Find the value of k that makes the above function a proper p.d.f. Part b) Find E(X).SHOW ONLY THE SET-UP FORMULA AND GRAPH IT7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]Determine the conditional probability distribution of Y given that X = 2. Where the joint probability density function is given by f(x,y)= 1 - (x + y) for 1 < x < 4 and 0 < y < 3.5. Determine the value of c that makes the function f (x, y) = c(x + y) a joint probability density function over the range 0 < x < 3 and 0 < y < x. Answer is C = 2/27 a. From the answer in Question 5, Determine the marginal probability distribution of X. b. From the answer in Question 5, Determine the conditional probability distribution of X given that Y = 2. c. From the answer in Question 5, Determine E(Y | X = 1). (up to 6 decimal place)Use R to complete the following. R code and R output are requested for solution. Please keep your solution clear and easy to read. Unclear solution will not be grade. O for manual solution. Carry out a simulation experiment to illustrate the central limit theorem when the population distribution is arcsine with probability density function f(x) = 1 T√x(1-x) x = (0, 1). Consider the four sample sizes n = 1, n = 3, n = 5, n = 10, and in each case use 10,000 replications. Note: To simulate a random sample of size n from the arcsine distribution in R, use the command rbeta(n,1/2,1/2).Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON