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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

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    $\begingroup$ 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
    Commented Aug 31, 2021 at 10:57
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    $\begingroup$ 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
    Commented Aug 31, 2021 at 11:13
  • $\begingroup$ @B.Liu Thanks for your reminder. Actually, it comes from Example 8.9 $\endgroup$
    – soplus2018
    Commented Aug 31, 2021 at 11:44
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    $\begingroup$ 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 ♦
    Commented 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). $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 31, 2021 at 14:14

1 Answer 1

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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.

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  • $\begingroup$ 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
    Commented Sep 1, 2021 at 23:12
  • $\begingroup$ en.wikipedia.org/wiki/Binomial_distribution $\endgroup$
    – Sycorax ♦
    Commented Sep 1, 2021 at 23:24

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