ZZa,Z,Z,be4sandard normal mandom variabls. Let Y Z+23+2 22 Deistmine dhe v so that PY Sk).025
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
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.
14. Let S be the cylinder {(x, y, z)| x2 +22 = 25,0 SY <2}. Orient S by the outward pointing normal n = (x/5,0,2/5). Evaluate Fds, where F(x, y, z) = (x – 2, y, x + 2). Hint: The area of a cylinder of radius r and height h is 2nrh.
Prove Valid: 1. (z)(Pz --> Qz) 2. (Ex) [(Oy • Py) --> (Qy • Ry)] 3. (x) (-Px v Ox) 4. (x) (Ox --> -Rx) ... :. (Ey) (-Py v -Oy) 1. (x) [(Fx v Hx) --> (Gx • Ax)] 2. -(x) (Ax • Gx) ..... :. (Ex) (-Hx v Ax) 1. (x) (Px --> [(Qx • Rx) v Sx)] 2. (y) [(Qy • Ry) --> - Py] 3. (x) (Tx --> -Sx) .... :. (y) (Py --> -Ty)
(a) Let X, have a chi-squared distribution with parameter V, and let X, be independent of X, and have a chi-squared distribution with parameter vz. Show that X, + X, has a chi-squared distribution with parameter v, + V Let Y = X1 + Xy. Identify the correct expression for Fly). Fyly) = (f1 +49 (0) BM={{1*(**) 2*3)**- _jei OFW - 1 -{{49)..:@) ib) dx1 FY) = -xq12 dx + -*2/20 1²ax 2 1/2 O Fy(y) :{"(****): 19). x 22...
#2 : Let X and Y be independent standard normal random variables, let Z have an arbitrary density function, and form Q = (X+ZY)/(V1+ Z2). Prove that Q also has a standard normal density function
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outward pointing normal in dS terms r and θ. This cone has an open bottom. . The integrand must be fully simplified » Do not evaluate the integral
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the...
4. Let f(x, y, z) = rytan'() + z sin(xy), < = wy=v²v, z = ". Find fu and , using the chain rule.
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...
1. Suppose that E(X) E(Y) E(Z) 2 Y and Z are independent, Cov(X, Y) V(X) V(Z) 4, V(Y) = 3 Let U X 3Y +Z and W = 2X + Y + Z 1, and Cov(X, Z) = -1 Compute E(U) and V (U) b. Compute Cov(U, W). а.