5.6.8. Suppose a random sample of size n is drawn from the pdf 1 fy(y; 0) = 0≤ y ≤ 0 Ꮎ Find a sufficient statistic for 0.

7.4.1

5.65. Find the mean µ, variance o², and standard deviation o of each distribution: 2 3 8 (a) (b) X f(x) X f(x) 1/4 -2 1/3 1/2 1/4 -1 7 1/2 1/6

16. Let X, X2 X, be a random sample from a normal distribution, X, N(6, 25), and denote by X and S2 the sample mean and sample variance. Use tables from Appendix C to find each of the following: (a) P[3 < X < T. (b) P[L860< 3(X-6)/S]. (c) PIS <319375]

Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter ß. Given Y = y, the number of accidents that the individual suffers in years 1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X₁, Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X₂ = k2 … . . Xn = kn) = [° ƒ (y, k1, k2, ... kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson distribution with parameter y.

2

An individual who has automobile insurance from acertain company is randomly selected. Let Y be thenumber of moving violations for which the individualwas cited during the last 3 years. The pmf of Y is y p(y) 0 0.6 0.25 0.10 0.05 2 4 6 Suppose an individual with Y violations incurs a surcharge of $100. Calculate the standard deviation of the surcharge Y. 1.48 1.72 2.2 1.2

6.2.8. Let X be N(0, 0), 0 <0 < x. (a) Find the Fisher information I(0). (b) If X₁, X2,..., Xn is a random sample from this distribution, show that the mle ofis an efficient estimator of 0. (c) What is the asymptotic distribution of √(-0)?

9.6 Suppose that Y₁, Y₂,... Y, denote a random sample of size n from a Poisson distribution (Y₁ + Y₂)/2 and 2₂ = Y. Derive the efficiency of 2₁ relative with mean λ. Consider ₁ = to Â₂.

4.5-1