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Probability And Statistical Inference (10th Edition)
- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0arrow_forwardIf a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise. Derive P.G.F of X and compute E(2x+1)arrow_forwardLet X be a continuous Random variable with probability density function f(x) = 2x , 0<x<1 0 , otherwise Find 1) P(X<=0.4) 2) P(X>3/4) 3) P(X>1/2) 4) P(1/2 <x < 3/4) 5) P(X>(3/4) / X>(1/2)) 6) P(X<3(/4) / X>(1/2))arrow_forward
- Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?arrow_forwardConsider a random variable, Y , which has a quasi-Bernoulli structure. With probability p ∈ [0, 1] it takes value 0. With probability (1 − p) it is described by a continuous random variable, X , with the following PDF, f_X (x)=1+x, x∈[−1,0), −1≤x<0f_X (x)=1−x, x∈[0,1], 0≤x≤1f_X (x)=0, otherwise , x>1 Obtain CDF of Y, F_Y (y), and draw the sketch.arrow_forwardThe joint probability function of two discrete random variables X and Y is given by Ax,y) = c(2x+y), where x and y can assume all integers such that 0< xarrow_forward
- Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z).arrow_forwardLet X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.arrow_forwardsuppose that the probability density function of x is f(x)={3x^2, 0, 0<x<1 elsewhere. determine p(x<1/3), P(1/3 <=x<2/3), and P(x=>2/3)arrow_forward
- Let X and Y be random variables, and a and b be constants. a) Prove that Cov(aX, bY) = ab Cov(X,Y). b) Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.arrow_forwardLet X, Y, and Z be random variables, and let Cov(⋅,⋅) denote the covariance operator as usual. Suppose that the variance of X is 0.7, Cov(X,Y) = 0.4, Cov(X,Z) = 1.2, and Cov(Y,Z) = 0.8. Find each of the following to two decimal places. (a) Cov(3Y, 3X) (b) Cov(3Y + 3, 3X + 8Z)arrow_forwardIf X is uniformly distributed over (1, 2), find z such that P ( X > z+μx ) = 1/4arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning