3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
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.
PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
Exercise 4.9. Let X ~ Poisson(10). (a) Find P(X>7). (b) Find P(X < 13 X > 7).
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...
5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.
3. (10 points) Let X be a continuous random variable with CDF for x < -1 Fx(x) = { } (x3 +1) for -1<x<1 for x > 1 and let Y = X5 a. (4 points) Find the CDF of Y. b. (3 points) Find the PDF of Y. c. (3 points) Find E[Y]
Let X be a random variable with the following cumulative distribution function (CDF): y<0 (a) What's P(X < 2)? (b) What's P(X > 2)? c)What's P(0.5 X < 2.5)? (d) What's P(X 1)? (e) Let q be a number such that F(0.6. What's q?