Let X~N(1,2) and Y~N(3,2). Define the distribution of 2(X-Y).
#2 2. Let X, N o ?) for i=1,2. Show that Y = X1 + X, and Z X; - X2 are independent. 3. Let 2-N(0,1) and W x (n) with Z be independent of W. Show that the distribution of T- tudiatvihustion with n deerees of freedom. (Hint: create a second variable U - find the joint distribution
2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution 2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution
Let X be distributed as N(0, 1). Define Xn (1)"X, n 1,2, a. [3 pts] Show that Xn-X. b. [3 pts] Show that Xn -» X
Let X be a uniform(0, 1) random variable and let Y be uniform(1,2) with X and Y being independent. Let U = X/Y and V = X. (a) Find the joint distribution of U and V . (b) Find the marginal distributions of U.
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
6. Let X have a N(1,2) distribution. Using only the tables, find: a) P(X 1.5) b) P(-1.1 X < 3.3) c) P(X-.9) d) A point c such that P(X > c) = 01 e) A point d such that P(X < d) 005
8. Let Xi be iid N(μ, σ2) random variables. Define Y-Σ, Xi-Find the distribution of Y. a.
DSP 4. (12 points) (a) (4 points) Let x[n] = {1,2, 1, 2} and h[n] = {1,-1,1, -1} be two length-4 sequences defined for 0 <n<3. Determine the circular convolution of length-4 y[n] = x[n] 4 hin). (b) (6 points) Find the 4-point discrete Fourier transform (DFT) X[k], H[k], and Y[k]. (c) (2 points) Find the 4-point inverse DFT (IDFT) of Z[k] = {X[k]H[k].
Let f(x,y)=x^2*y. Find the directional derivative of f at (1,2) in the direction of (3,4).
Question: Let Y, Y be a random sample of size n=2 from a distribution with Pdf f(y; 6) = (6)e-0 OLGLD and o elsewhere We reject Ho : O=2 and accept H1:0=1 if the observed values of Y la are such that: fu, ; 2) fűz; 2) at f(4,;1) fŲz;1) 1 * Find the Significance level and the Power of the test when Ho is falso; given that {0:0=1,2}.