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1. Suppose E[X]-2 and E Y]-1. Evaluate the followings: a) E3X ] b) EX 2Y] c)
Suppose that EX-EY-0, var(X) = var(Y) = 1, and corr(X,Y) = 0.5. (i) Compute E3X -2Y]; and (ii) var(3X - 2Y) (ii) Compute E[X2]
Suppose that f (x II 2y), 0 < x < 1,0 < y < 1. Find EX + Y).
Evaluate the general solutions of the following differential equations: y"-2y'+y = ex/x
2y - 1 E Evaluate by + 2 if possible, for a) y = 2 and b) y = -4. Express answers as integers or simplified fractions, if necessary. 2y - 1 a) For y = 2, the expression 6y + 2 is b) For y = -4, the expression 2y - 1 6y + 2 is
4. Suppose Σί, 1 Xi-1, ΣΙ-, x-2, and n-5. Evaluate the followings: a) Σί-1[Xi + X) b) 5. Let X-1 Σ., x1-1 with n-100. (i) Obtain ΣΙ.i Xu (ii) Evaluate Σ.1x,-X) and X +1
Find the area of the region between the curves y=e3x andy=e−x from x=−1 to x=1 Flhd the area of the region between the curves! ve mul v=e* from 3-10 = 1 y = €3x and y=e- from x = -1 to x = 1 Area = | 21.678
a. Show = b. Show E{(X-EX)(Y-EY)}=E{(X-EX)Y}
Given the following equations y´(x)+y(x)=e-x; 2y´(x)+y(x)=e-x ; y´(x)+2y(x)=e-x a. Use the integration factor and find your solutions b. Discuss the trend behavior of the solutions and (x) as follows: 1. compare the graphs of y (x) with the graphs of F (x) 2. Compare the graphs of y (x) with respect to the coefficients of the differential equations
2. Evaluate the line integral / (x+2y)dx + r’dy, where C consists of the path C from (0,0) to (3,0), the path C2 from (3,0) to (2,1), and the path C3 from (2,1) to (0,0) by applying the following steps. (a) Evaluate (x + 2y) dx + c'dy, by parametrizing C C (b) Evaluate [ (x + 2y)dx + x>dy, by parametrizing C, (c) Evaluate | (x + 2y)dx + x’dy, by parametrizing C3 (d) Evaluate (+2y)dx + xºdy
1. Suppose the joint density of X and Y is given by f(x,y) = 6e-3x-2y, if 0 < x < inf., 0 < y < inf, 0 elsewhere. Part A, Find P( X < 2Y) Part B, Find Cov(X,Y) Part C, Suppose X and Y have joint density given by f(x,y) = 24xy, when 0<= x <=1, 0 <= y <=1, 0 <= x+y <=1, and 0 elsewhere. Are X and Y independent or dependent random variables? why?