The joint probability density is for X and Y are f(x, y). Give the exact answer. f(x,y)= { 2 (* +y) 0 < x < 1, 0
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- Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), f(x₁, x₂) = x1 > 0, x₂ > 0 otherwise. Q2(i.) Sketch(Shade) the support of (X1, X₂). Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint…9 Let the probability density of the two-dimensional random variable (ξ, η) be (4e**, x > 0; f;(x)=• (0, 6e", y>0; -6y f,(v) =; 0, x<0. y<0. Find E(2 { +3 n ).D is defined in the other picture atatched
- A number x is selected at random from [4, 20]. The probability density function for x is given by f(x) = for 4 ≤ x ≤ 20. 2 Find the probability that a number selected is in the sub-interval [7, 18].The probability density of a random variable is (k> 0,0 < x <∞0). f(x) = ax²e-kx Find (a) the coefficient a, (b) the distribution function of the random variable X, (c) the probability that the random variable belongs to the interval (0, 1/k).نقطتان )2) Q10) If you know that is the following function is joint probability density function, * ?what is the value of t S9xy 0ASAPEstimate f(x) dx given that f is a continuous function containing (-2, 8), (–1,0), (4, 125) , and points of the form (x, e") for all positive integers x using an appropriate Newton-Cotes quadrature rule utilizing all the given data in this problem.Let X will be continuous random variable with probability density function as below. f(x) = {4x , (0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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