# Estimators

#### alkiii321

##### New member
We make subsequent throws of a fake cubic cube for which the probability of falling out six is 1/6 - epsilon, the probability of falling out of one is 1/6 + epsilon and the others eyes drop out with the same probability 1/6. Provide a consistent and unbalanced estimator epsilon parameter and calculate its variance. I need help i start my trip with this ad it is so hard for me :/

#### Romsek

##### Full Member
$$\displaystyle \text{Let }\bar{x} \text{ be the vector of }n \text{ rolls}$$
$$\displaystyle \text{Let }o = \text{#1's in }\bar{x},~s=\text{#6's in }\bar{x}$$
$$\displaystyle p_\epsilon(\bar{x}) = \left(\dfrac 1 6\right)^{n-o-s}\left(\dfrac 1 6 - \epsilon \right)^s \left(\dfrac 1 6 + \epsilon\right)^o$$

$$\displaystyle \log(p_\epsilon(\bar{x}) = (n-o-s)\log\left(\dfrac 1 6\right) + s \log\left(\dfrac 1 6 - \epsilon\right) + o \log\left(\dfrac 1 6 + \epsilon\right)$$

The maximum likelihood estimate will result from minimizing this expression over $$\displaystyle \epsilon$$

This is done using the usual calculus minimization technique of setting the first derivative to zero and solving.

Can you finish?

#### alkiii321

##### New member
I don't understand. What is an estimator?

#### Romsek

##### Full Member
I don't understand. What is an estimator?
If you don't know what an estimator is why have you been given this problem? (Maybe your school uses a different name.)

An estimator is a function that takes a set of observations and reduces them to an estimate of some property of the underlying random variable.