**Normal distribution**

The Normal distribution is depicted in ** image0 **with mean 0, the center of the curve, and different kinds of standard deviation that modify the shape of the bell. For more information about this distribution, look at chapter 2.2.3 Normal Distributions & Normality

In ** image0.1, **you can look at the probability formula of the normal distribution.

**Lognormal distribution**

The Lognormal distribution is often used to count the time duration, for example, measuring the time where the process is down. If you look at * image5*, you have multiple plots of the lognormal distribution, and it always has a positive skew, which means that the tail is on the right of the image. Like the normal distribution, the mean and the standard deviation define the aspect of the curve.

In ** image5.1**, you can look at the probability formula of the lognormal distribution.

Exponential Distribution

The Exponential Distribution is used to describe the time between two events in a process where each event is indipents and occurs at a costant avarage rate. In the ** image8 **you can look the exponential distribution with different rate.

Example: How much time pass before a component breakdown if we have a rate of 0.5? In the, on the x-axis, you have the amount of time; more is the time, less is the probability.image8

In image8.1 you can look at the formula of the probability distribution.

where:

- Lambda is the rate of the events that occurs.

**Chi-Square distribution**

The Chi-Square distribution* *with K degree of freedom (df) is the sum of the squares of K independent standard normal random variables. It can be used to test a population’s variance against a known conflict or to look if an observed distribution fits a theoretical one. In ** image7, **we look at the chi-square distribution with different degrees of freedom.

**We will look more about this distribution in the chapter about the Hypothesis Test.**

**F Distribution**

The F distribution** **is used to test the variance from two normal populations independent of each other. You can look at the probability distribution representation in ** image6**.

The F distribution is also related to the chi-square distribution representing the ratio of two chi-square distributions with two different degrees of freedom (df1 and df2), as shown in ** image6.1**.

where:

**df**is calculated with n_{1}_{1}-1, where n_{1 }is the number of samples in X_{1}**df**is calculated with n_{2}_{2}-1, where n_{2}is the number of samples in X_{2}

**This formula isn’t the probability formula; it only looks at how the distribution is composed.We will look more about this distribution in the chapter about the Hypothesis Test.**

T-student distribution

The T-Student distribution is used when you’re estimating the mean of a normally distributed population in situations where the sample size is small (n<30) and the population’s standard deviation is unknown. In the ** image9.1 **you can look how the T-Student is composed:

where:

**df**is the degree of freedom;**Z**and**K**are two independent variables, where Z is a normal standard distribution and K is a chi-squared distribution with df degree of freedom.

** This formula isn’t the probability formula; it only looks at how the distribution is composed.**

**We will look more about this distribution in the chapter about the Hypothesis Test.** ** **

The degree of freedom (df) is the number of samples less than one. In the ** image9, **you can look at the plot of the t-student distribution at different df values.