I am studying probability and maximum likelihood estimation. From one website that I read, they use an example about blue and red balls probability as below
I have a bag that contains 3 balls. Each ball is either red or blue, but I have no information in addition to this. Thus, the number of blue balls, call it θ , might be 0 , 1 , 2 , or 3 . I am allowed to choose 4 balls at random from the bag with replacement. We define the random variables X1 , X2 , X3 , and X4 as follows
Xi = 1 if the ith chosen ball is blue and 0 if the ith chosen ball is red
Note that Xi's are i.i.d. and Xi∼Bernoulli(θ/3)
My question is what does Bernoulli(θ/3) means? and how do we get the value θ/3?
Here is the website that I read: https://www.probabilitycourse.com/chapter8/8_2_3_max_likelihood_estimation.php
Thank you
You have $\theta$ blue balls out of a total of $3$ balls. If you take one ball randomly, the probability to get a blue ball is $\theta/3$. That is the argument for the Bernoulli in your post.
Let's say after sampling 4 times, you get $bbrb$. The probability to get this result is given by the following expression:
$$ \frac{\theta}{3} \cdot \frac{\theta}{3} \cdot \frac{3-\theta}{3} \cdot \frac{\theta}{3} $$
Now to answer your other question, namely, how to obtain $\theta/3$ or equivalently, how to obtain $\theta$? The answer is, you need to choose among $0,1,2,3$ such the the previous expression has a maximum value. Hence the name maximum likelihood estimation. In this case, $\theta=2$ gives us the maximum, so we estimate that there are two blue balls in the bag.