A random sample of size n = 2 is taken from the p.d.f
f(x) = {1 for 0 ≤ x ≤ 1 and 0 otherwise.
Find P(X-bar ≥ 0.9)
random sample of size n from the p.d.f. 1.8 On the basis of a (x,θ)-θΧθ-1 , 0 < x < 1, θ E Ω = (0,0)derive the MLE of θ
(1 point) Let X1 and X2 be a random sample of size n= 2 from the exponential distribution with p.d.f. f(x) = 4e - 4x 0 < x < 0. Find the following: a) P(0.5 < X1 < 1.1,0.3 < X2 < 1.7) = b) E(X1(X2 – 0.5)2) =
1. Consider a random sample of size n from a population with pdf: f (x) = (1 -p-p, 0 <p<1, x= 1, 2, ... (a) Show that converges in probability to p. (b) Show that converges in probability to p (1 – p). (c) Find the limiting distribution of
2. Let Xi,... ,Xn be a random sample from a distribution with p.d.f for 0 < x < θ f(x; 0) - 0 elsewhere . (a) Find an estimator for θ using the method of moments. (b) Find the variance of your estimator in (a).
1. Suppose that the p.d.f. of a random variable X is as follows: for 0<x<2, for 0 〈 x 〈 2. r for 0<< f(x) = 0 otherwise. Let Y - X (2 - X). First determine the c.d.f. of Y, then find its p.d.f. (Hint: when computing c.d.f., plotting the function Y- X(2 - X) which may help. )
2.a. Let X1, X2, ..., X., be a random sample from a distribution with p.d.f. (39) f( 0) = (1 - 1) if 0 < x <1 elsewhere ( 1 2.) = where 8 > 0. Find a sufficient statistic for 0. Justify your answer! Hint: (2(1-)). b. Let X1, X2,..., X, be a random sample from a distribution with p.d.f. (1:0) = 22/ if 0 < I< elsewhere where 8 >0. Find a sufficient statistic for 8. Justify your...
QUESTION 12 Let the random variable X and Y have the joint p.d.f. f(x,y) =(zy for 0< <2, 0 < y <2, and z<y otherwise Find P(0KY <1) 16 QUESTION 13 R eter to question 12. Find P(o < x <3I Y-1).
5. A sample of size 2 is taken from the density function f(x)-1, 0sxs 1, and zero elsewhere. What is the probability that X is at least.9? 6. A sample ofsze 2 is taken from probability function P(X=1) = p-1-P( X = 0), 0<p<1. i) Find P(X Find P(S22.5). [Hint: For a sample ofsze n-2, s-= Σ (Xi-X) 2/(2-1)-(X-%)2/21 2 i- 1
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
7. (1 point) Let X be the mean of a random sample of size 36 from the uniform distribution U(7,15) Find P(11.3 <X < 11.5)