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

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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:

P(k, \lambda) = \frac{\lambda^ke^{-\lambda}}{k!}

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?

P(3,2) = \frac{2^3\times e^{-2}}{3!} = 0.180

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

\sum_{k=0}^{4} \frac{2^k\times e^{-2}}{k!} = 0.94734825712

Normal distribution

The probability density of normal distribution is:

N( \mu , \sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}

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 $x$

\begin{align*} \Phi(x) &= \frac{1}{2}(1+ error(\frac{x-\mu}{\sigma\sqrt{2}})) \\ error(z) &= \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-x^2} dx \end{align*}

Practice

Hackerrank has some exercises for you to test your knowledge:

Next lesson: Central limit theorem