10. (8 marks) Suppose Y, Y is a random sample of independent and identically distributed random...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
3. (20 marks) Suppose Y...Y is a random sample of independent and identically distributed Gamma(c, B) random variables. Suppose c is a known constant. a) (5 marks) Find an exact (I-a)100% CI. forty: cß based on Y. Hint: Make use of the Chi-Square distribution when finding your pivot. b) (5 marks) Find an approximate (1-α)100% CI. forMy-cß based on only Y using a n (Y-CB) ~ Normal (0, Normal approximation and the pivot Z- c) (5 marks) Find an approximate...
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...
If X and Y are independent and identically distributed uniform
random variables on (0,1) compute the joint density of
U = X+Y, V = X/(X+Y)
Part A,
The state space of (U,V) i.e. the domain D over which
fU,Y (u,v) is non-zero can be expressed as
(D = {(u,v)
R x R] 0 < h1(u,v) < 1, 0 < h2(u,v)
< 1} where x = h1 (u,v) and y = h2
(u,v)
Find h1(u,v) = (write a function in terms...
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
Problem 8: 10 points Suppose that (X, Y) are two independent identically distributed random variables with the density function defined as f (x) λ exp (-Ar) , for x > 0. For the ratio, z-y, find the cumulative distribution function and density function.
. A random sample of size n is taken from a population that has a distri- bution with density function given by 0, elsewhere Find the likelihood function L(n v.. V ) -Using the factorization criterion, find a sufficient statistic for θ. Give your functions g(u, 0) and h(i, v2.. . n) - Use the fact that the mean of a random variable with distribution function above is to find the method of moment's estimator for θ. Explain how you...
Suppose Y1, Y2, …, Yn are independent and identically distributed random variables from a uniform distribution on [0,k]. a. Determine the density of Y(n) = max(Y1, Y2, …, Yn). b. Compute the bias of the estimator k = Y(n) for estimating k.
2. (15pts) Let X1, X2 be independent and identically distributed with Uniform(0,) density. (a) Is Y-X1 + X2 a sufficient statistic for θ? Hint: You need to find the conditional density of (X1, X2) given Y = X1 + X2. (b) Consider now S := max(X1, X2). 1s S a sufficient statistics for θ?