# Ten days of statistics (6) - Poisson & normal distribution

## Poisson experiment

Poisson experiment is a statistical experiment that has the following properties:

- The outcome of each trial is either success or failure.
- The average number of successes $\lambda$ that occurs in a specified region is known.
- The probability that a success will occur is proportional to the size of the region.
- The probability that a success will occur in an extremely small region is virtually zero.

## Poisson distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution:

Where

- $k$ is the number of expected successes
- $\lambda$ is the average number of successes
- $e$ is Euler’s number, $e = 2.71828$

### Example

Vova sells 2 cars per day on average. What is the probability of him selling 3 cars today?

What is the probability of him selling at most 4 cars today?

## Normal distribution

The probability density of normal distribution is:

Where $\mu$ is the mean, $\sigma$ is the standard deviation

### Why is it called normal?

Because apparently it is the **most popular** distribution found in natural (learn more)

## Cumulative probability

Let $\Phi(x)$ is the cumulative distribution function of $x$, denotes the probability of all values less than or equal to $x$

## Practice

Hackerrank has some exercises for you to test your knowledge:

- https://www.hackerrank.com/challenges/s10-poisson-distribution-1/problem
- https://www.hackerrank.com/challenges/s10-poisson-distribution-2/problem
- https://www.hackerrank.com/challenges/s10-normal-distribution-1/problem
- https://www.hackerrank.com/challenges/s10-normal-distribution-2/problem

Next lesson: Central limit theorem