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

image0 - normal distribution
image0 – normal distribution

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

image01 - Normal distribution formula
image01 – Normal distribution formula

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.

image5 - Lognormal distribution
image5 – Lognormal distribution

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

image5.1 - Longnormal distribution formula
image5.1 – Longnormal distribution formula


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.

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

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

image8.1 Exponential distribution formula
image8.1 Exponential distribution formula

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.

image7 - chi square distribution
image7 – chi square distribution

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.

Image6 - F distribution
Image6 – F distribution

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.

image6.1 - F-Distribution
image6.1 – F-Distribution

where:

  • df1 is calculated with n1-1, where n1 is the number of samples in X1
  • df2 is calculated with n2-1, where n2is the number of samples in X2

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:

image9.1 - t-student
image9.1 – t-student

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.

image9 - T-Student distribution
image9 – T-Student distribution
Share on: