Question

5.1 Refer to Exercise 1.6, and derive the moment estimate of θ. Also, compare it with the MLE θ= 1/x. 1.6 If the independent r.v.'s. Xi, .., X" have the Geometric p.d.f.

Answer 1

p.d.f.of Geometric Distribution is given by:

,

for x = 1, 2,...

The mean of geometric distribution is given by:

-e+29(1-0)+36(1-0)2 +

,

since

Thus, we get:

(1)

Given:

Independent r.v.'s: X₁, X_2,...,X_n.

So,

Mean is given by:

$E(X)=ar{X}=rac{sum_{i=1}^{n}X_{i}}{n}$ (2)

Equating (1) & (2), we get:

Thus, the moment estimate of is given by:

$θ=$.

This is same as MLE of

$θ=$