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Are the following functions concave, convex, or neither for x > 0? (i) f(z) =ztt convex,...
across Let F = (x i+yj +z k'). What is the outward flux of F (x2 + y2 +22)8 a sphere of radius R> 0 centered at the origin?
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z) : x2 + y2 = p2, z = h}. 1 Show that for any r > 0 and h ER, Sony F. dx = 0
3. For each of the following distributions, find the expected value of X (a) f(z,8) = θ(1-9)"-, z = 1, 2, . . . . 0 (b) f(z,8) = (θ + 1)2 , z > 1,0 > 0 (e) f(z,8) = θ2ze-e", z > 0, θ > 0 9s 1 -0-2
2. For the difference cquation, X2+] = ax, + b = f(x,), where 0 <a < 1 and b> 0, use the solution given in (1.12) to find the following limit: lim ->XX7. Show that this limit is also a fixed point of the difference cquation, that is, it is a solution x of t = f(x) (see Figure 1.2).
-3x > 0 An exponential distribution is given by f(x) zero, x <0 Find the distribution of the random variable Y X2
(b) ONLY! Though you can use the result from (a) without proof (a) Let F(x) = x + x2 + x3 +... and let G(x) = x - x2 + x3 – x4.... Show that for k > 1 and n>k, (4")F(x)* = (n = 1) and if n < k then [x"]F(x)k = 0. (b) Show that G(F(x)) = x.
For what values of x is the function f(x) = x3 + 15 x2 + 63 x increasing? f(x) is increasing when z 〈 and when x>
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x, 0) =1 + x2 using the method of characteristics. Find the u(x, y). Substitute your found solution u(x, y) in the equation and verify that it satisfies the equation. solution explicitly in the form u =