Binomial Distribution

 Binomial Distribution:

• Definition: The Binomial Distribution is a discrete probability distribution that models the number of successes (usually denoted as $x$) in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes - success (usually denoted as 1) and failure (usually denoted as 0). It is named because it deals with "bi" or two outcomes.

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  Probability Mass Function (PMF): The PMF of the Binomial Distribution is defined as:

  $P(X=x)=\left(\genfrac{}{}{0px}{}{n}{x}\right)\cdot p^x\cdot (1-p{)}^{n-x}$

  Where:

  • $X$ is a random variable representing the number of successes.
  • $x$ is the specific number of successes.
  • $n$ is the total number of trials.
  • $p$ is the probability of success in each trial.
  • $\left(\genfrac{}{}{0px}{}{n}{x}\right)$ represents the binomial coefficient, which counts the number of ways to choose $x$ successes out of $n$ trials.

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  Mean and Variance: The mean (expected value) of the Binomial Distribution is $np$, and the variance is $np(1-p)$.

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  Graphical Representation:

  Here's a bar graph illustrating the Binomial Distribution for a specific example:

  

  In this graph, you can see the probability of getting a specific number of successes ($x$) in a fixed number of trials ($n$) with a given probability of success ($p$). Each bar represents a different value of $x$.

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  Use Cases:

  • The Binomial Distribution is commonly used in scenarios where there are a fixed number of trials, each with two possible outcomes, and you want to know the probability of getting a certain number of successes. Typical use cases include:
    • Coin flips (number of heads in $n$ flips).
    • Pass/fail experiments (number of passes in $n$ attempts).
    • Election predictions (number of votes for a candidate in $n$ districts).

  It's also used for hypothesis testing, where you want to determine if an observed outcome is significantly different from what you would expect by random chance.

The Binomial Distribution is a fundamental distribution in probability theory and statistics, widely applicable in various fields, including biology, finance, and quality control.