# Maximum Likelihood Estimator, Single Observation

1. Mar 10, 2012

### bitty

1. The problem statement, all variables and given/known data
An observation X has density function: f(x,/theta)=6x/(t^3)*(t-x) where t is a parameter: 0<x<t.

Given the single observation X, determine the maximum likelihood estimator for t.

2. Relevant equations
Included below

3. The attempt at a solutionFor a sample size of n, the likelihood function is
L(t)=product[6x_i/(t^3)*(t-x_i)] from i=1 to n.
To maximize a product, we take the log L[t] and differentiate with respect to t and equate to 0.

d/dt [Log[t]=3/t+Sum[1/(t-x_i)] for i=1 to n.
However, I don't know how to solve 3/t+Sum[1/(t-x_i)]=0 for t since there is a t on the denominator in the sum. Is there a way to do this?

Or have I misinterpreted the question? Since the question says, "Given a single observation X" am I not supposed to take the product over n samples but rather set n=1 when finding my likelihood function, which would then be L(t)=6x/(t^3)*(t-x) without a product?

*I accidentally posted this in the incorrect section (pre-Calc) so I apologize for the double post.

2. Mar 11, 2012

### sunjin09

It sounds like you don't need the sum, i.e., n=1 indeed