Calculates the **probability** or the **percentile** or the **probability between two values.**

Normal distribution, Binomial distribution, T distribution, F distribution, Chi-square distribution, Poisson distribution, Weibull distribution, Exponential distribution.

Choose 𝑥

For example the bell curve line represent the density of the normal distribution.

The area under the density for a specific X range, the integral of the range, is the probability to get value in this range. p(X=𝑥) = 0.

This is generally because many natural processes are naturally distributed or have a very similar spread.

Some examples of normally distributed data include height, weight, and error in measurements.

The Normal distribution has a symmetric "Bell Curve" structure. more data exist around the center, which is the average, and as further the value is from the center the less likely it occurs.

Usually, when adding independent random variables, the result tends toward the normal distribution (CLT - The Central Limit Theorem)

You can calculate the values of any normal distribution based on the standard normal distribution (a normal distribution with mean equals zero and standard deviation equals one)

when X distributes normally, μ mean and σ standard deviation, Z=(x-μ)/σ distributes as the standard normal distribution, so you can calculate any normal distribution based on the standard normal distribution.

PDF(𝑥) = | 1 | exp(-^{} | (𝑥 - μ)^{2} | ) |

σ√(2π) | 2σ^{2} |

Z = | 𝑥 - μ |

σ |

When calculating the percentile, there is usually no X that meets the exact probability you enter. The tool will calculate the X that will generate distribution which is equal or bigger than the input probability but will calculate the probabilities for both X and X-1.

When the tool can't calculate the distribution or the density using the binomial distribution, due to a large sample size and/or a large number of successes, it will use the

P(X=𝑥) = ( | 𝑥 | )p^{𝑥}q^{n-𝑥} |

n |

Z = | 𝑥 - np |

√(np(1 - p)) |

T distribution looks similar to the normal distribution but lower in the middle and with thicker tails. The shape depends on the degrees of freedom, number of independent observations, usually number of observations minus one (n-1). The higher the degree of freedom the more it resembles the normal distribution.

PDF = | 1 | (1+ | 𝑥^{2} | )^{-(k+1)/2} |

B(1/2,k/2)√k | k |

Let Z

Let X= [Z

PDF(𝑥, k) = | 1 | x^{k/2-1}e^{-𝑥/2} |

2^{k/2}Γ(k/2) |

Let Z

Let X

Let Z'

Let X

Let X = | X_{1}/n |

X_{2}/m |

X

X

Example of use: ANOVA test, F test for variances comparison.

λ is the average number of events per unit of time.

The number of events on

The time between events distributes

𝑥 ≥ 0

P(X=𝑥) = | λ^{x}e^{-λ} |

𝑥! |

A unique character of the distribution is

Memorylessness example: If the probability for burned-out bulb event in the next 2 months is 0.3, and you waited 1 year without any event, now the probability for an event in the next 2 months is still 0.3.

λ - duration between the events

When using λ be sure to check if it is duration between events or rate - events per unit of time, since some people uses λ as duration and some uses λ as rate:

𝑥 ≥ 0

P(X=𝑥) = | e^{-𝑥/λ} |

λ |

Example: When the event is a faulty lamp, and the average lamps that need to be replaced in a month are 16.

The number of lamps that need to be replaced in 5 months distributes Pois(80). since: 5 * 16 = 80.

The time between faulty lamp evets distributes Exp(1/16). The unit is months.

When k = 1 (shape), the failure rate is constant. This is the

When k > 1 (shape), the failure rate increases over time.