3. Suppose that y E C" is defined so that y(k-1 for 0 k-1 and y(k-0...
4. Let {Sn,n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y, max{Sk, 1 3 k S nt, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain
Problem 3 A discrete random variable Y takes values {k= 0, 1, 2, ...,} such that PLY Z k} = ()* for k 20. 1. Derive P[Y = k) for any k > 0. 2. Evaluate expectation, E[Y] = 3. Given E[Y(Y - 1)] = 15 , find variance of Y, Var[Y] =
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint pdf given by fx,YZ(xgz) = k xyz if 0 S$ 1,0 rS 1,0 25 1 ) and fxyZ(x,y,z) = 0, otherwise. (a) Find k so that fxyz(x.yz) is a genuine probability density function. (b) Are X,Y,Z independent? (c) Find PXs 1/2, Y s 1/3, Z s1/4). (d) Find the marginal pdf fxy(x.y). (e) Find the marginal pdf fx(x). Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint...
te tt 9 Let S be the surface defined by r2 y-22= 1 and 0 15 points by the normal direction toward the z-axis. Find the flux of the velocity field V 1and oriented (z2-ry2)i+(2.2y -yz2)j+ (y2z- 2r2)k across S Solution. To use Gauss Theorem, define C and C2 such that C1 {(a, y, z ) | a + y? < 2, z = 1} C2={(r,y,)| +y1.0} 11 TALK X W P S ww F6 & # 2 3 4...
using discrete structures 3. Consider the function F(x, y, z) for x, y, z z 0 defined as follows: a. F(x, y, 0)-y+1 b. F(x, 0, 1)-x c, F(x, 0, 2) = 0 d. F(x, 0, z+ 3)-1 e. F(x, y, z)-F(x, F(x, y-1, z), z-1) Using Induction, prove the following a. F(x, y, 1)-x +y b, F(x, y, 2) = xy c. F(x, y, 3)-xy 3. Consider the function F(x, y, z) for x, y, z z 0 defined...
(c) A digital filter has transfer function 1 Н(2) z 1/2 Evaluate the response function of the filter, Y(z)= X(z)H(z), for the sequence (i 2* x(n)a. (Use the geometric series 1-c k 0 (ii By using partial fractions, determine the response of the filter, y(n), to the input x(п) %— а". (iii What is the response to the input data x(n) (1)"? [Note: the Z- transform of a sequence x(n) is defined as X(z) x(n)z. The n-0 inverse Z- transform...
1. Consider the function h:Z+ +Z+ defined by h(n) = l{k e Z+ : k|n}l. The bars around the set mean that we are taking the size of the set. Thus h(n) is the number of positive divisors of n. (a) Make a table of values for h(n) for 1 sn < 10. Write one or two sentences describing how you found the values in the table. (b) Find the value of h(90). Explain how you found your answer. (c)...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
4. Let {Sn, n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y max{Sk, 1 Sk n, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain.
2 Suppose Vf(x, y, z) - 2xyze i ze* j + ye k. If f(0, 0, 0) 1, find f(2, 2, 3).