8. Let X.(i-12) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2尸/( X2-X)2 < c ) =.90 b. Find P(2 X1 -3 X21.5) c. Find 95th percentile of the distribution of Y-2X -3X2
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
Find all integers x, y, 0 < x, y < n, that satisfy each of the following pairs of congruences. If no solutions exist, explain why. (a) x + 5y = 3(mod n), and 4x + y = 1(mod n), for n = 8. (b) 7x + 2y = 3(mod n), and 9x + 4y = 6(mod n), for n=5.
Let x[n] and y[n] be periodic signals with common period N, and let z[n] = { x[r]y[n – r) r=<N> be their period convolution. Let z[n] = sin(7") and y[n] = { . 0 <n<3 4 <n <7 Asns? be two signals that are periodic with period 8. Find the Fourier series representation for the periodic convolution of these signals.
8. Let X (i-1,2) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2 )2/( X2 -X1)2 < c ) =.90 b. Find P(2 X1 -3 X2< 1.5) c. Find 95th percentile of the distribution of Y-2 X1 -3 X2
Exercise 6. Let (X, Y) be uniform on the unit ball, i.e. it has density if 2+y2 < 1, if 2y1 -{i fx.y) (r, y) Find the density of X2Y
In the exercise below, let U = {x|XE N and x < 10} A = {xx is an odd natural number and x < 10} B = {x x is an even natural number and x < 10} C = {x|XE N and 3 <x<5} Find the set. ВПС {4} {2, 4, 6, 8, 10) {1,2,3,4,5,6,7,8,9) {1,2,3,4,5,6,7,8,9,10)
Let Xi, , X. .., Exp(β) be IID. Let Y max(Xi, , h} Find the probability density function of Y. İlint: Y < y if and only if XS for i 1,,n.
real analysis questions Find the interior of the following sets. (1): {1/n: neN}: (2): (0,5) (5, 7); (3): {re Q:0<r <2}. Classify each of the following sets as open, closed, or neither. (1): {: | - 51 < 1}; (2): {x: (x-3) > 1}; (3): {:13 -4)<4}.
Exercise 6.14 Let y be distributed Bernoulli P(y = 1) unknown 0<p<1 p and P(y = 0) = 1-p f or Some (a) Show that p E( (b) Write down the natural moment estimator p of . (c) Find var (p) (d) Find the asymptotic distribution of vn (-p) as no. as n> OO.