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Binomial Distribution

Ankit0

The binomial distribution is a probability distribution that describes the number of successes in a sequence of n independent trials, each of which has a probability of success.

Assumptions:

  • The trials are independent.
  • Each trial has a constant probability of success p.
  • The number of trials n is finite.

Formula:

The probability of getting exactly k successes in n trials is given by the following formula:

P(X = k) = n choose k * p^k * (1-p)^(n-k)

where:

  • P(X = k) is the probability of getting exactly k successes
  • n is the number of trials
  • k is the number of successes
  • p is the probability of success
  • n choose k is the binomial coefficient, given by the formula:

n choose k = n! / (k! * (n-k)! )

Moments:

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

E = np

The variance of a binomial distribution is given by:

Var = np(1-p)

Applications:

The binomial distribution is used to model a wide variety of phenomena, including:

  • The number of defective items in a batch of products
  • The number of heads in a coin toss
  • The number of successes in a raffle

Example:

Suppose you have a coin with a probability of heads of 0.4 and you toss it 10 times. The number of heads that you will get in 10 tosses will follow a binomial distribution with n = 10 and p = 0.4. The probability of getting exactly 3 heads is given by:

P(X = 3) = 10 choose 3 * 0.4^3 * (1-0.4)^(10-3) = 0.216

This means that there is a 21.6% chance of getting exactly 3 heads in 10 tosses of a coin with a probability of heads of 0.4.

FAQs

  1. How do you identify a binomial distribution?

    A distribution can be identified as binomial if it involves a fixed number of trials, each with two possible outcomes. The probability of success must remain constant throughout the trials, and the trials must be independent of one another.

  2. What is the difference between Bernoulli and binomial distribution?

    A Bernoulli distribution deals with a single trial that has two outcomes, such as success or failure. In contrast, a binomial distribution involves multiple independent Bernoulli trials and calculates the probability of a certain number of successes within those trials.

  3. What is the relationship between binomial and normal distribution?

    As the number of trials in a binomial distribution increases and the probability of success is not extremely close to 0 or 1, the binomial distribution begins to resemble a normal distribution. This is known as the normal approximation to the binomial distribution and is often used to simplify calculations for large sample sizes.

  4. When should I use Bernoulli distribution?

    The Bernoulli distribution is appropriate when analyzing a single trial with a binary outcome, such as determining the result of a single coin toss (heads or tails) or checking if a product passes or fails a quality test.

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