2. Let f(z) be the principal branch of i.е., f(z) exp@ Log(z)}. Co mpute (e)dz where...
5. Let f(x) = e-} Log(z) (that is, f is the principal branch of z-1/2). Compute [flade, where (a) (2 points) y is the upper half of the unit circle C(0) from +1 to -1; (b) (2 points) y is the lower half of the unit circle C1(0) from +1 to -1.
10. Let R > 0, Describe the arc「with parametrisation z(t)-Reit for-π/2-t-π/2 Use th is parametrisation to calculate the integral log z dz, where log z denotes the principal value of the logarithm.
-wa exp{-(20 )2}, where The Normal(μ,02) distribution has density f(x) -oo < μ < oo and σ > 0. Let the randon variable T be such that X-log(T) is Normal(μ, σ2). Find the density of T. This distribution is known as the log normal Do not forget to indicate where the density of T is non-zero. 10.
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
Let X1, X2,.. Xn be a random sample from a distribution with probability density function f(z | θ) = (g2 + θ) 2,0-1(1-2), 0<x<1.0>0 obtain a method of moments estimator for θ, θ. Calculate an estimate using this estimator when x! = 0.50. r2 = 0.75, хз = 0.85, x4= 0.25.
29. Let Z be a standard normal random variable. (a) Compute the probability F(a) = P(2? < a) in terms of the distribution function of Z. (b) Differentiating in a, show that Z2 has Gamma distribution with parameters α and θ = 2.
3. Let X1 , X2, . . . , Xn be a randon sample from the distribution with pdf f(r;0) = (1/2)e-z-8,-X < < oo,-oc < θ < oo. Find the maximum likelihood estimator of θ.
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C?
complex analysis Let f(z) be continuous on S where for some real numbers 0< a < b. Define max(Re(z)Im(z and suppose f(z) dz = 0 S, for all r E (a, b). Prove or disprove that f(z) is holomorphic on S.
Let F(x, y, z) = 4i – 3j + 5k and S be the surface defined by z = x2 + y2 and x2 + y2 < 4. Evaluate SJ, F.nds, where n is the upward unit normal vector.