0
$\begingroup$

Assume a bag that contains 3 balls. Each ball is either red or blue. The number of blue balls, call it θ, might be 0, 1, 2, or 3. Choose 4 balls at random from the bag with replacement. define the i.i.d. random variables $X_1$, $X_2$, $X_3$, and $X_4$ as follows

\begin{equation} X_i = \left\{ \begin{array}{l l} 1 & \qquad \text{if the $i$th chosen ball is blue} \\ & \qquad \\ 0 & \qquad \text{if the $i$th chosen ball is red} \end{array} \right. \end{equation}

Given $X_i \sim Bernoulli(\frac{\theta}{3})$. I understand $(x_1, x_2, x_3, x_4)=(1,0,1,1)$ means there are 4 observed values, blue for the 1st observation, red for the 2nd observation, and blue for both 3rd and 4th observation.

Given $X_i \sim Binomial(3, \theta)$ What could $(x_1,x_2,x_3,x_4)=(1,3,2,2)$ mean?

The question comes from Example 8.8 in this statistics tutorial

$\endgroup$
6
  • 1
    $\begingroup$ If $x_i$ is Bernoulli, then $(x_1,x_2,x_3,x_4)=(1,3,2,2)$ doesn't mean anything, it couldn't possibly attain those values. Now, if $x_i$ is Binomial, it makes sense. $\endgroup$
    – Firebug
    Aug 31, 2021 at 10:57
  • 3
    $\begingroup$ Are you asking for an interpretation on what $(X_1, X_2, X_3, X_4) = (1, 3, 2, 2)$ could mean given $X_i \sim Bin(\theta)$, as stated in Example 8.8? The section preceding Example 8.8 generalizes the concept from a Bernoulli distributed example, and Example 8.8 is no longer related to the example you have quoted in your question. $\endgroup$
    – B.Liu
    Aug 31, 2021 at 11:13
  • $\begingroup$ @B.Liu Thanks for your reminder. Actually, it comes from Example 8.9 $\endgroup$
    – soplus2018
    Aug 31, 2021 at 11:44
  • 2
    $\begingroup$ It means that in the first experiment, you observed 1 blue ball. In the second experiment, you observed 3 blue balls. In the third and fourth experiments, you observed 2 blue balls. Does this answer your question? Or are you asking something else? $\endgroup$
    – Sycorax
    Aug 31, 2021 at 11:50
  • $\begingroup$ Is this a question from a course or textbook? If so, please add the self-study tag & read its [wiki] (stats.stackexchange.com/tags/self-study/info). $\endgroup$ Aug 31, 2021 at 14:14

1 Answer 1

1
$\begingroup$

It means that in the first experiment, you observed 1 blue ball. In the second experiment, you observed 3 blue balls. In the third and fourth experiments, you observed 2 blue balls.

$\endgroup$
2
  • $\begingroup$ Thank you so much. Does 3 in $Binomial(3, \theta)$ mean there are 3 balls in the bag? Note that in the second experiment, 3 blue balls are observed which means there already are 3 blue balls in the bag. If I need to ask this in a new post, please let me know. $\endgroup$
    – soplus2018
    Sep 1, 2021 at 23:12
  • $\begingroup$ en.wikipedia.org/wiki/Binomial_distribution $\endgroup$
    – Sycorax
    Sep 1, 2021 at 23:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.