Suppose that UCC is open and connected and a E U. Let F:= {f € H(U)|...
Suppose that UCC is open and connected and a E U. Let F:= {f € H(U)| Re(f)> 0, f(a) 1} Show that Fis normal.
Suppose that UCC is open and connected and a E U. Let F:= {f € H(U)| Re(f)> 0, f(a) 1} Show that Fis normal.
Prove or Disprove:
Let p E P(F) and suppose that deg p > 1 and p is irreducible. Then p(a)メ0 for all a E F.
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
4. Find d > 0 such that d 1000, 5 | d, d| 60, and d/2 | 75
Find the probability that Y is greater than 3.
Let Y have the probability density function f(y) = 2/y3 if y> 1, f(y) = 0 elsewhere.
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
PROB5
Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise. and the p.d.f'of2 is Seu, v>0, fv (v otherwise. Let X = V2V cos U and Y = 2V sin U. Show that X and Y are independent standard normal variables N(0,1).
1 Let X1,..., Xn be iid with PDF x/e f(x;0) ',X>0 o (a) Find the method of moments estimator of e. (b) Find the maximum likelihood estimator of O (c) Is the maximum likelihood estimator of efficient?
Assume U U(0,1), meaning that U is a continuous random variable, uniformly distributed in the interval (0, 1). Fix λ > 0 and define X =ナIn U. What is the density of X?
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