New answers tagged binomial-distribution
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Avoid unnecessary calculations when multiplying matrices if only need one element of resulting matrix
Paul Sinclair found that the bottom left element (hereinafter $e_{10}$) is a sum of products of $x_i$s where the first and last index of each term are of opposite parity to $N$ and no two adjacent ...
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Existence and Uniqueness of MLE for a collection of Bernoulli Random Variables
You don't need to compute any derivatives to show the existence and uniqueness of the optimum; instead, a much simpler strategy is to show that $\log p(Y, \theta)$ is strictly concave, which implies ...
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Avoid unnecessary calculations when multiplying matrices if only need one element of resulting matrix
Let $Q := \{1,\dots,N\}$ be the set of all indices, and for any $S \subseteq Q$, understand $\prod_{i \in S} \mathbf M_i$ to refer specifically to the product in increasing order of indices: ${\bf M}_{...
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Binomial distribution as sum of Bernoulis - Independent?
Pairwise independence is not sufficient to conclude a sum of Bernoulli random variables is binomially distributed.
Here is an example with $21$ random variables, each with probability $1/6$. You have ...
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Beta-Bernoulli conjugate prior Bayes risk
Let's use $S = \sum\limits_{i=1}^nX_i$ to save typing. Then
$W$ has a $\operatorname{Beta}(\alpha,\beta)$ prior distribution with mean $\mathbb E[W]=\frac{\alpha}{\alpha+\beta}$ and variance $\...
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How many times should you play to be sure of winning and losing at least once?
The chance of winning them all is $p^n$ and the chance of losing them all is $(1-p)^n$. As these are mutually exclusive the chance of either one is the sum, $p^n+(1-p)^n$. It is this you want ...
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How to bet on individual games in best of seven series so that no matter how my team wins I make same profit? (Assume games are coin flips)
This can be solved with backwards induction and is usually easiest to see when you draw out the tree of possible outcomes.
Let's denote the current state of the game by $S_{i,j}$ where $i$ is the ...
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Generalizing Bernoulli to Binomial without summing
Your joint distribution of i.i.d. categorical random variables $(C_1,\ldots,C_n)$ really takes values in $S^n$ where $S$ is the set of categories. Enumerate $S = \{ s_1,\ldots,s_k \}$, then for $1\le ...
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