What is Bernoulli distribution?

A Bernoulli distribution is a discrete probability distribution that describes the probability of achieving a “success” or “failure” from a Bernoulli trial. A Bernoulli trial is an event that has only two possible outcomes, which we typically call “success” and “failure.” For example, a coin toss is a Bernoulli trial, because the only possible outcomes are heads (success) and tails (failure).

The probability of success

The probability of success in a Bernoulli trial is denoted by the letter p. The probability of failure is 1 — p. For example, if the probability of flipping a heads is 0.5, then the probability of flipping a tails is 0.5.

The Bernoulli probability mass function

The Bernoulli probability mass function (PMF) is a function that gives the probability of getting a “success” in a Bernoulli trial. The PMF is denoted by f(x), where x is the number of successes. For example, if x = 1, then f(x) is the probability of getting one success in a single Bernoulli trial.

The Bernoulli PMF is:

f(x) = p^x (1 - p)^(1 - x)

where:

• x is the number of successes
• p is the probability of success
• (1 — p) is the probability of failure

Example

Let’s say we flip a coin 10 times. We want to know the probability of getting 5 heads. The number of successes in this case is x = 5. The probability of success is p = 0.5, because the probability of flipping a heads is 50%. The probability of failure is 1 — p = 0.5.

The Bernoulli PMF for this example is:

f(5) = (0.5)^5 (0.5)^5 = 3125/1024

This means that the probability of getting 5 heads in 10 coin flips is 3125/1024, which is about 30.77%.

Bernoulli distribution in other applications

Bernoulli distribution is a simple but powerful tool that can be used to model many different types of events. For example, it can be used to model the success or failure of a medical treatment, the probability of a customer making a purchase, or the likelihood of a student passing an exam.

Bernoulli distribution is also a building block for more complex probability distributions, such as the binomial distribution and the Poisson distribution. These distributions are used to model events that have more than two possible outcomes.

I hope this explanation of Bernoulli distribution is clear and helpful. Please let me know if you have any other questions.

Happy Learning!!!!!

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