2. Let (X,Y) be a continuous bivariate random variable having the joint probability density function f(x,y) = cxy, 0≤x≤y≤2 for some real constant c. (e) Compute μX,μY ,σX2 ,σY2 ,Cov(X,Y), and ρ. (f) Find g(y|x = 12), the conditional probability density function of Y given X = 12.
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2. Let (X,Y) be a continuous bivariate random variable having the joint probability density
(f) Find g(y|x = 12), the conditional probability density function of Y given X = 12.
(g) Find P(Y > 34|X = 210
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- The life lengths of two transistors in an electronic circuit is a random vector (X; Y ) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y ) is given by 2e-(x+2y) X> 0, γ> 0 fx,ylx,v) = { else Then the probability that the first transistor last for at least half hour given that the second one lasts at least half hour equals Select one: a. 0.7772 b. 0.3935 10 c. 0.606 d. 0.6318 e. 0.3669a) Let X be a continuous random variable with the following probability density function (2x3 f(x)={11 2 +5) , 0Let X and Y be continuous random variables with joint probability density function f(x, y) = (3/200y if 0 < 5x < y < 10 O otherwise Find Cov(X, Y)Explain A, B, C7. 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.]Let X₁, X₂,..., X, be independent, uniformly distributed random variables on the interval [0, b]. a) Find the probability distribution function of X(n) = max(X₁, X₂,..., Xn). Fx (t) = for 0 ≤t≤ b and zero elsewhere I b) Find the probability density function of X(n)- fx (t) = c) Find the expected value of X(n) E(X(n)) = for 0 ≤t≤ b and zero elsewhere2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0Let (X; Y ) be a continuous random vector with joint probability density function 0.5 -1Consider the probability density fx (x) = a. eb x! where X is random variable whose allowable values range from x = -o to + o. Find (a) the CDF (b) the relation between a and b (c) the probability that x lies between 1 and 2.Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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