Problem 1: Show E[Y-G(X)|2] is minimized using G(X) = EX ing G(X) EYAX
a. Show = b. Show E{(X-EX)(Y-EY)}=E{(X-EX)Y}
4. (a) For the random variable X, show that E[(x - a)?] is minimized when a = E(X). (b) For random variables X and Y, show that Var(X+Y) S Var(X) + Var(Y), that is, the standard deviation of the sum is less than or equal to the sum of standard deviations. (c) For random variables X and Y, prove the Cauchy-Schwartz Inequality: [E(XY)]2 < E(X) E(Y2)
(a) For the random variable X, show that E[(x – a)?] is minimized when a = E(X). (b) For random variables X and Y, show that Var(X + Y) < Var(x) + Var(Y), that is, the standard deviation of the sum is less than or equal to the sum of standard deviations. (c) For random variables X and Y, prove the Cauchy-Schwartz Inequality: [E(XY)]? 5 E(X2) E(Y2)
4. (a) For the random variable X, show that E[(x -a)?] is minimized when a = E(X). (b) For random variables X and Y, show that Var(X+Y) S Var(X) + Var(Y), that is, the standard deviation of the sum is less than or equal to the sum of standard deviations. (c) For random variables X and Y, prove the Cauchy-Schwartz Inequality: [E(XY)] = E(X) E(Y)
7. Show that E(X-EX]) (Y-E[Y)] = E[XY]-EXEY]
let g=(x e R:x>1) be the set of all real numbers greater than 1. for X,Y e G, define x * y=xy - x-y +2. 1. show that the operation * is closed on G. 2. show that the associative law holds for *. 3.show that 2 is the identity element for the operation *. 4. show for each element a e G there exists an inverse a-1 e G.
Let XN(0, 1) and Y eX. (X) (a) Find E[Y] and V(Y). (b) Compute the approximate values of E[Y] and V(Y) using E(X)(u)+"()VX) and V((X))b(u)2V(X). Do you expect good approximations? Justify your an- Swer
Let XN(0, 1) and Y eX. (X) (a) Find E[Y] and V(Y). (b) Compute the approximate values of E[Y] and V(Y) using E(X)(u)+"()VX) and V((X))b(u)2V(X). Do you expect good approximations? Justify your an- Swer
1. Find 8 different 2-level minimized circuits to realize each of the following functions. 1. F(W,X,Y,Z) = {m (2,4,6,7,12,14,15) 2. G(W,X,Y,Z) = (x + Y' + Z) (X' + Y + Z) W • Using algebraic techniques • Using network conversion
2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G (0,y) = 6,(1, y) = 0 does not exist, but a modified Green's function Ĝ(x,y) satisfying a2G 22 = (x - y) -1, G.(0,y)=G.(1,y) = 0 does. How would you use G to solve problem (1) when f satisfies the condition that you found for a solution to exist? Hint: is f(x) = f(u) (8(x - y) - 1) dy?
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol, we see that f(n)- f(n-1)+f(n-1) hence we can define f as f-{(my) є N N : (n-ОЛ у-1) v (n > 0A3y, E N(n-1, y') є f Лу = y, +y®)), and show that this is a function by induction.)
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol,...