c) Let Y be Poisson(A). Apply continuity theorem to show that Y Z~N (0, 1) Y-A...
1. Let (N(t))>o be a Poisson process with rate X, and let Y1,Y2, ... bei.i.d. random variables. Fur- ther suppose that (N(t))=>0 and (Y)>1 are independent. Define the compound Poisson process N(t) Y. X(t) = Recall that the moment generating function of a random variable X is defined by ºx(u) = E[c"X]. Suppose that oy, (u) < for all u CR (for simplicity). (a) Show that for all u ER, ºx() (u) = exp (Atløy, (u) - 1)). (b) Instead...
problems binomial random, veriable has the moment generating function, y(t)=E eux 1. A nd+ 1-p)n. Show that EIX|-np and Var(X) np(1-p) using that EIX)-v(0) nd E.X2 =ψ (0). 2. Lex X be uniformly distributed over (a b). Show that ElXI 쌓 and Var(X) = (b and second moments of this random variable where the pdf of X is (x)N of a continuous randonn variable is defined as E[X"-广.nf(z)dz. )a using the first Note that the nth moment 3. Show that...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Let Z ~ N(0,1) and let Y = Z2. Find the distribution of Y. Hint: Use moment generating function. Let X ~ N(j = 1, 02 = 4). If Y = 0.5*, find E(Y?). Hint: Use moment generating function.
Finish the proof of Theorem 3.14. Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
Answer C 6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] = ax +b and Var[Y|X =x] = 1 a) compute E[Y] b) compute Var[Y] c) Find E[XY]
9. Let Q be the solid bounded by the cylinder x2 + y2 = 1 and the planes z = 0 and z = 1 . Use the Divergence Theorem to calculate | | F . N dS where s is the surface of Q and F(x, y, z) = xi + yj + zk. (a) 67T (d) 0 (b) 1 (e) None of these (c) 3π 9. Let Q be the solid bounded by the cylinder x2 + y2...
Prove that if X 20, Y 2 0 and 0 p1, then E(X +Y)] Show that for any real numbers x > 0 and y > 0, E(X)E(YP). HINT: Here is how you can show the above formula holds. Start off by letting 0y. Use the fact that the function g(z) - z is concave-down (i.e., "spills water") on (0, oo) and is thus bounded above by its tangent line at any particular point. Find the tanget line at the...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...