j: ~ A. J.l Jfl it~#* -'1f ~ ffl 1 0 5 ~ 1¥- & -'.A ± JJI fl. t«. t«."!

(1)

j: ~ A. J.l Jfl it~#* -'1f ~ ffl 1 0 5 ~ 1¥- & -'.A ± JJI fl. t«. t«."!

#

^{§ :}

#lifs.

^jJt

tt

2015. 10. 23

1. (8%) (7%) Let f(t) and F(t) stand for the respective probability density function and cumulative distribution function of a discrete non-negative random variable T with the sup- port {tt < ... < tm}, S(t)

=

^1-F(t), and Ai

=

P(T =tilT~ ti), ⁱ

=

^{1, .}^..^,m. Express f(ti) and S(ti) in terms of A/s.

2. (10%) Let X have a Gamma distribution with parameters a

>

1 and (3. Compute the mean of the random quantity

1 /

^X^.

3. (15%) Let Xt, ... , Xn be a random sample from a continuous distribution F(x) with the corresponding order statistics Xc1), . . . , X(n)· Derive the distribution of F(X(i)), i = 1, ... , n.

4. (10%) (15%) Let X1 , ..• , Xn be a random sample from N(J-L, a²^). Find the maximum likelihood estimator of~( (x -~J.) /a), where ~( ·) represents the cumulative distribution function of a standard normal random variable and x is a given value, and derive its asymptotic distribution.

5. (8%) (7%) Let X_{1 ,}... , Xn be a random sample from Poisson(A) and A have a Gamma( a, !3) distribution. Find the posterior distribution of A and the Bayes estimator of A under the absolute error loss function.

6. Let X₁, . . . , Xn be a random sample from a density function f(xiA) = Oe->.x J{(O,oo)}(x)

with A> 0.

(6a) (10%) Show that the rejection region of a likelihood ratio test of Ho : A ⁼ Ao versus HA : A =I= Ao is of the form {(Xt, ... , Xn) : xe-AoX}, where

X

is the sample mean of

(6b) (10%) Find a valid p-value for the above hypotheses.