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**Table of contents**

- references - Recommended books on experiment design? - Cross Validated
- Design of Experiments (DOE)
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Two random variables x and y are independent , if their probability distributions can be expressed as a product of two factors, one involving only x and other involving only y :. In other words, occurrence of one event does not affect occurrence of another event in any way. Two random variables x and y are conditionally independent given a random variable z if the conditional probability distribution over x and y factorizes in the way for every value of z:.

The expectation or expected value of some function f x with respect to a probability distribution P x is the average or mean value, that f takes on when x is drawn from P. For discrete variables this can be computed with a summation as:. The variance gives a measure of how much the values of a function of a random variable x vary as we sample different values of x from its probability distribution:.

The covariance gives some sense of how much two values are linearly related to each other, as well as the scale of these variables:. Several simple probability distributions are useful in many contexts in machine learning. Its name might sound too complex and scary but it is the easiest distribution to understand. All the cricket fans knows that, at the beginning of a cricket match how do we decide who is going to bat or ball?

A toss!

It all depends on whether you win the toss or lose the toss. A Bernoulli distribution has only two possible outcomes namely 1 success and 0 failure , and a single trial. So the random variable x which has Bernoulli distribution can take value 1 with the probability of success say p and value 0 with the probability of failure q or 1-p. The probability mass function PMF for this is given by:.

The probabilities of success and failure need not to be equally likely. The expected value of a random variable x from a Bernoulli distribution can be computed as:. The multinoulli or categorical distribution is a distribution over a single discrete variable with k different states, where k is finite. In Bernoulli distribution, random variable has two possible outcomes where a categorical random variable has multiple possible outcomes.

When there is a single trial, categorical distribution is known as multinomial distribution. The possible outcomes the sample space are 1,2,3,4,5,6. When you roll a die the outcomes are 1 to 6. The probabilities of getting these outcomes are equally likely and that is the basis of uniform distribution. In uniform distribution, all the n number of possible outcomes are equally likely. The PDF for this can be given by:.

The mean and variance for a random variable x following a uniform distribution is:.

## references - Recommended books on experiment design? - Cross Validated

Binomial distribution has only two possible outcomes 1 success and 0 failure , but unlike Bernoulli distribution it is repeated multiple times or have multiple trials. We can use the above cricket example again here, for a toss you either get heads success or tails failure. On the basis of the above explanation, the properties of a Binomial Distribution are. The mean and variance of a binomial distribution are given by:. This is the most commonly normally! Any distribution is normal distribution if it has the following characteristics:. The PDF of a random variable x following a normal distribution is given by:.

In the absence of prior knowledge about what form a distribution over the real numbers should take, the normal distribution is a good default choice for two major reasons:. The Central Limit Theorem CLT is a statistical theory states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

The amazing and very interesting intuitive thing about the central limit theorem is that no matter what the shape of the original parent distribution, the sampling distribution of the mean approaches a normal distribution. A normal distribution is approached very quickly as n increases, Note that n is the sample size for each mean and not the number of samples. This distribution can be applied to some interesting examples that you can relate very easily.

Suppose you work at a call centre, how many calls do you get in a day? It can be any number. Now the entire number of calls at a call centre in a day can be modelled by Poisson Distribution.

Some more examples are:. Poisson distribution is applicable in a situations where events occur at random point of time and space wherein our interest lies only in the number of occurrences of the event. The following assumptions are necessary for Poisson distributions:. The PMF of a random variable x for Poisson distribution is given by:. The graph of a Poisson distribution is shown below:. The mean and variance for Poisson distribution:. What about the interval of time between the call?

These type of questions where we need to find waiting time for a given event to occur, can be answered by exponential distribution. It is widely used for survival analysis.

The PDF for a random variable x to have exponential distribution is given by:. Mean and variance for Exponential distribution is:. Also, the greater the rate, the faster the curve drops and the lower the rate, flatter the curve. This is explained better with the graph shown below. Laplace distribution represents the distribution of differences between two independent variables having identical exponential distributions.

It is also called double exponential distribution.

## Design of Experiments (DOE)

Like normal distribution, this distribution is unimodal one peak and symmetrical. However, it has a sharper peak than normal distribution. The general PDF for laplace distribution is:. It is also common to define probability distributions by combining other simpler probability distributions. Mixture distribution is a mixture of two or more probability distributions. The parent populations can be univariate or multivariate, although the mixed distributions should have the same dimensionality.

In addition, they should either be all discrete probability distributions or all continuous probability distributions. A mixture distribution can be defined by the following formula:. Examples when to use mixture distribution:. A very powerful and common type of mixture model is the Gaussian mixture model, in which the components are Gaussians. Each component has separate parametrized mean and covariance. This is a very wide topic and I am going to discuss it in detail in later article.

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This concludes this part of the series. Here I covered all the basics about probability, probability distribution and other related topics. In the next article we will see some more broad and complicated topics that I am currently working and soon upload them. I hope you enjoy this part and we will learn more in the upcoming part too. Marginal probability explanation. Introduction to Conditional Probability and Bayes theorem by analytics vidhya. Bayes Rule and conditional probability example. Article on Bayesian Probability. Explained pdf on conditional independence. Article on Basic Probability Distributions.

Common types of Probability Distributions. Article on Probability Distributions by Sean Owen.

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Article on Categorical Distribution. Article on Mixture Distribution. Detailed article on Mixture Distribution. Stackoverflow answer on Why is Normal Distribution a default choice? Usefully, statsmodels has classes to perform a power analysis with other statistical tests, such as the F-test, Z-test, and the Chi-Squared test. In this tutorial, you discovered the statistical power of a hypothesis test and how to calculate power analyses and power curves as part of experimental design.

Do you have any questions? Ask your questions in the comments below and I will do my best to answer. It provides self-study tutorials on topics like: Hypothesis Tests, Correlation, Nonparametric Stats, Resampling , and much more Great article, thanks for the attacking this important, often neglected topic, from a machine learning perspective. Maybe the use of the phrase significance level is misleading.

My question is — knowing the typical standard deviation of the type of experiment I am running, can I tailor my power analysis to my specifics to get a more accurate idea of the samples size I will need?

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As it is, this seems to be a one-size-fits-all test. The standard deviation is part of the effect size. If you know the expected mean difference between your populations and the standard deviation you should be able to calculate the effect size for your specific experiment. Excellent Blog. I love reading this and gives me great information. Thanks for giving such a great information to us….!!!! This is awesome, thank you for being so methodical and providing a context-rich explanation to power analyses with Python, it helps a bunch!