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


  • kk is the number of expected successes
  • λ\lambda is the average number of successes
  • ee is Euler’s number, e=2.71828e = 2.71828


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

P(3,2)=23×e23!=0.180P(3,2) = \frac{2^3\times e^{-2}}{3!} = 0.180

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

k=042k×e2k!=0.94734825712\sum_{k=0}^{4} \frac{2^k\times e^{-2}}{k!} = 0.94734825712

Normal distribution

The probability density of normal distribution is:

N(μ,σ)=1σ2πe(xμ)2/(2σ2)N( \mu , \sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}

Where μ\mu is the mean, σ\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 Φ(x)\Phi(x) is the cumulative distribution function of xx, denotes the probability of all values less than or equal to xx

Φ(x)=12(1+error(xμσ2))error(z)=2π0zex2dx\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*}


Hackerrank has some exercises for you to test your knowledge:

Next lesson: Central limit theorem

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