Total marks 144.Let X and Y be random variables on a probability space (N, F, P)that take values in [0, ∞). Assume that the joint density function of Xand Y on [0, ∞) × [0, ∞) is given byf(x, y) = 2e-2x-yFind the probability P(0 ≤ X ≤ 1,0 ≤ y ≤ 2).(ii)spectively.[6 Marks]Find the the probability density function of X and Y, re-[5 Marks]111)Are the X and Y independent? Justify your answer![3 Marks]

(i) Find P(0≤X≤1 , 0≤Y≤2):The probability is computed by integrating the joint PDF over the given…

Answered: Total marks 14 4. Let X and Y be random…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.3: The Addition And Subtraction Formulas

Problem 76E

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Let X be a uniformly distributed continuous random variable that can take values in the interval (-1,1). Y Let Y=1-X2 be defined as a function of the random variable X. a) Mathematically find the probability density function (PDF) of Y. b) Find mathematically the covariance, Cov[X,Y], of X and Y. X and Y are uncorrelated (uncorrelated)? Are X and Y independent? c) With the help of the "rand" command in MATLAB, the random variable X is randomized into 1,000,000 random variables. and the corresponding Y=1-X2 values. Plot the histogram of Y. The histogram you found in (a) Does it match the PDF? Comment. d) Estimate Cov[X,Y] using the 1,000,000 X and Y values you generated in MATLAB obtain For this, from the definition Cov[X,Y] = E[(X-E[X])(Y-E[Y])] and the "mean" command Does it match the value you found in (b)? Comment.
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Q2 Let (X1, X₂) be jointly continuous with joint probability density function e-(x1+x2), 0 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). 1 = Q2 (iv.) Let Mx, (t) = Mx₂ (t) (1 t), for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. Q2(v.) Let Y₂ = X₁ — X₂, and Mx₁ (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…
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