If X ~ N(?, ?2), what is the median of X? Justify your answer (this does...
3. If X ~ Nga ?2), what is the median of X? Justify your answer (this does not have to be a proof). 4. If Y ~ Gamma(?, ?) , show that (a) E(Y)-??. (b) E(Y2)-?2(z + 1)a (c) V(Y)-??2
5. Determine whether the following statements are True or False. Justify your answer with a proof or a counterexample as appropriate. (a) The relation S on R given by xSy if and only if X – Y E R – N is an equivalence relation.
(b) (5 points) Are X and Y independent? Justify your answer. (c) (5 points) Identify the random variable expressed as the conditional expectation of Y given X, i.e., E[Y |X]. 6. (15 points) Consider two independent Bernoulli r.v., U and V, both with probability of success 1/2. Let X = U+V and Y = |U - VI. (a) (5 points) Calculate the covariance of X and Y , oxy.
1. Answer the following questions. Justify your answers. (a) (3 marks) For what values of x does the series 1 + 22x 2 + 24x 4 + 26x 6 · · · + 22nx 2n + · · · converge? (b) (9 marks) Is the following series absolutely convergent, conditionally convergent or divergent? i. X∞ n=1 2 √ n − 2 √ n + 1 ii. X∞ n=1 arctan(n) n2 + 1 iii. X∞ k=1 (−1)k k 1. Answer the...
Consider the discrete-time system with input x[n] and output y[n] described by : y[n]=x[n]u[2-n] Which of the following properties does this system possess? Justify your answer in each case. Do not use Laplace transforms a) Memoryless b)Time-invariant c) Linear d)Casual e) Stable
Does theta(n^3+2n+1) = theta(n^3) hold? Justify your answer.
I. (a) Compute g(x) = £ (n2 +n)x”. (b) Compute E (n?+n)/2". Justify your method. n=0 n=0
8. (10 Pts) Answer by True / False and justify your answer. (a) Let A be a 2 × 2 matrix such that(A2-Nthen, if A ±1 A--. (b) If C is a skew-symmetric matrix of odd order n, then |C-0 (c) If A is a square matrix, and the linear transformation L(z) Az is one-to-one, then the linear transformation x ? At is also one-toone. z), ? O (z, y, z) = (az, ay, 0), then V is not a...
7. Assume X gammala, y) and Y-gamma(8,7) are independent. (a) Show that U = X + Y gamma(a +.). (b) Show that V = X/(X+Y) beta(a. 8). (c) Show that U and V are independent. (d) Show that W = 72 gamma(1/2, 1/2) if Z N (0,1).
Problem 1. Consider the two data collections (x,..-, X) and (y,. ya). The median difference statistic is the median of the (v-x) differences for the mn possible (x, y.) pairings, i= 1, ..., mandj = 1, . . ., n. Construct two specific data collections (x,. Xm) and (y,. .. V) that demonstrate that the median difference statistic is not equal to the difference in the separate medians for the two collections Problem 2. A Venn diagram is a graphical...