Binomial Distribution

Binomial Random Variable

Definition

Recall a Bernoulli Random Variable is defined over a sample space of binary outcomes, a success s that occurs with probability \(p\) of success and a failure f that occurs with probability \(1-p\),

\[\begin{split}Y = \begin{array}{ c l } 1 & \quad \textrm{with probability} p \\ 0 & \quad \textrm{with probability } 1 - p \end{array}\end{split}\]

Consider a random variable defined as the sum of \(n\) Bernoulli random variables, \(Y_i\)

\[X = Y_1 + Y_2 + ... + Y_{n-1} + Y_n\]

Where each \(Y_i\) takes the value 1 with probability \(p\) or it takes the value 0 with probabilitiy \(1 - p\)

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From Conditional Probability, the probability of an intersection of independent events is the product of individual probabilitiy,

\[P(A \cap B) = P(A) \cdot P(B)\]

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Conditions

In order for an experiment to be Binomial, the experiment must the conditions just discussed. The summary below provides a list of each condition.

Binomial Conditions

1. The number of trials \(n\) must be fixed.

2. Each trial must be independent of the others.

3. Each trial must have a binary outcome, usually denoted success or failure.

4. The probability of success is the same in each trial.

Parameters

The Binomial Distribution has two parameters.

Binomial Parameters

1. \(n\): The number of trials.

2. \(p\): The probability of success in a single trial.

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Binomial Probability

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Probability Density Function

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\[p(x; n, p) = C^{n}_x \cdot p^{x} \cdot (1 - p)^{n-x}\]

Cumulative Distribution Function

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By definition,

\[F(x; n, p) = \sum^{x}_{i=0} C^{n}_i \cdot p^{i} \cdot (1 - p)^{n-i}\]

Expectation

TODO: derive through rules of independent random variable sums

Expectation of Binomial Random Variable

If \(\mathcal{X}\) is the number of successes in n independent trials, each with probability p, then the expectation of \(\mathcal{X}\), \(E(\mathcal{X})\), is given by,

\[E(\mathcal{X}) = n \ cdot p\]

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Standard Deviation

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Variance of Binomial Random Variable

If \(\mathcal{X}\) is the number of successes in n independent trials, each with probability p, then the variance of \(\mathcal{X}\), \(Var(\mathcal{X})\), is given by,

\[Var(\mathcal{X}) = n \ cdot p \cdot (1 - p)\]

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Standard Deviation of Binomial Random Variable

If \(\mathcal{X}\) is the number of successes in n independent trials, each with probability p, then the standard deviation of \(\mathcal{X}\), \(\sigma_{Bin(n,p)}\), is given by,

\[\sigma_{Bin(n,p)} = n \ cdot p\]

TODO: derive through rules of independent random variable sums