Suppose x1 9 x2 7 is a set of two independent observations

Suppose x1 = .9, x2 = .7 is a set of two independent observations from a population with pdf f(x; theta) = (theta + 1)x^(theta) for 0 <= x <= 1 and 0 otherwise.

a) What is the lilelihood function L(theta)?

b) Use the likelihood function to find the MLE for theta.

Solution

a) Given by: f(x;? ) = (?+ 1)x^(?) for 0 <= x <= 1 and 0 otherwise.

  We write the likelihood function as L(?;x)=? n i=1 f(X i ;?) or sometimes just L(?). Algebraically, the likelihood       L(? ; x) is just the same as the distribution f(x ; ?)

b)

Suppose that an experiment consists of n = 2 independent Bernoulli trials, each having probability of success p. Let X be the total number of successes in the trials, so that X?Bin(2,p) . If the outcome is X = 3, the likelihood is

L(p;x) =n!x!(n?x)! p x (1?p) n?x =2!3!(5?3)! p 3 (1?p) 5?3 ?p 3 (1?p) 2    like wize we need to do for both x1,x2.