CH4 - Binomial Distribution

Binomial Distribution

Summary

The binomial distribution models the probability of a specific number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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Definition

A binomial distribution describes the number of successes in a sequence of $n$ independent trials, where each trial has two possible outcomes: success or failure. The probability of success in each trial is denoted by $p$.

The binomial probability mass function (PMF) is given by:

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

where:

• $n$ is the number of trials,
• $k$ is the number of successes,
• $p$ is the probability of success on a single trial,
• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials.

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Properties

• Mean: The expected value (mean) of a binomial distribution is given by:

  $$\mu =np$$

• Variance: The variance of a binomial distribution is:

  $${\sigma }^2=np(1-p)$$

• Support: The possible values of $X$ range from $0$ to $n$, where $X$ is the number of successes.

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Example

Example: Tossing a Coin

Suppose you flip a fair coin ($p=0.5$) 10 times. What is the probability of getting exactly 6 heads?

Using the binomial formula:

$$P(X=6)=\left(\genfrac{}{}{0ex}{}{10}{6}\right)(0.5{)}^6(0.5{)}^4=\frac{10!}{6!(10-6)!}(0.5{)}^{10}$$

This simplifies to:

$$P(X=6)=210\cdot (0.5{)}^{10}=0.205$$

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Key Points

Key Characteristics of Binomial Distribution

• Trials are independent.
• Each trial has exactly two outcomes.
• Probability of success remains constant across trials.

Common Use Cases

• Quality control: Checking defective products in a batch.
• Clinical trials: Success rate of a treatment.
• Survey analysis: Probability of a certain number of people agreeing with a statement.

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Cumulative Binomial Probability

In practice, it is often useful to compute the probability of $X$ being less than or equal to a certain value. The cumulative distribution function (CDF) for the binomial distribution is:

$$P(X\le k)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{n}{i}\right)p^i(1-p{)}^{n-i}$$

This can be used, for example, to determine the likelihood of obtaining up to a certain number of successes in $n$ trials.

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Inverse Binomial Distribution

An inverse binomial distribution (or negative binomial distribution) models the number of failures needed before achieving $r$ successes, where $r$ is a fixed number of successes. Its probability mass function is:

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{k+r-1}{k}\right)p^r(1-p{)}^k$$

Here, $k$ represents the number of failures before the $r^{th}$ success.

Conceptual Question

How many trials are needed, on average, to achieve $r$ successes if the probability of success on each trial is $p$?

The expected value for the total number of trials is:

$$\mathbb{E}[T]=\frac{r}{p}$$

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Practical Considerations

• The binomial distribution assumes independence between trials. If the trials are dependent (e.g., without replacement), consider using a hypergeometric distribution.

• For large $n$ and small $p$, the Poisson distribution can approximate the binomial distribution when $\lambda =np$.

• The normal distribution can approximate the binomial distribution when $n$ is large, using the Central Limit Theorem:

  $$Z=\frac{X-np}{\sqrt{np(1-p)}}$$

This approximation is particularly useful for computationally expensive problems.