Probability distributions

Many natural phenomena or industrial processes depend on variables that take on unpredictable but completely random values. An example could be the value obtained following the throwing of a dice or the ambient temperature measured on the same day of the year at the same time. These variables are called random variables or causal (or stochastic) variables. Theorising a phenomenon of any nature that depends on random variables requires the introduction of the concept of probability distributions. In this blog we will go through the most common types of distribution and we will analyse them and their applications with real cases. First of all, we have to understand what is a probability distribution. A probability distribution is a mathematical model that relates values to the probability that the random variable can assume them. Probability distributions are used for modelling the behaviour of a phenomenon of interest in relation to the reference population, or the totality of cases for which the investigator observes a datum sample. In this context the variable of interest is seen as one random variable (or random variable, v.a.) whose law probability expresses the degree of uncertainty with which his values can be observed.

Discrete and continuous distributions

Based on the measurement scale of the variable of interest X, we can distinguish two types of probability distributions:

• continuous distributions: the variable is expressed on a continuous scale (ex: the diameter of the piston).
• discrete distributions: the variable is measured with integer numeric values (ex: number of elements not compliant or defective in a printed circuit board).
  Formally, probability distributions come expressed by a mathematical law called the function of probability density (denoted by f (x)) or function of
  probability (denoted by p (x)) for le respectively continuous or discrete destruction.

The mean (or expected value) µ and the variance σ 2 (deviation standard σ) of a v.a. X are the major parameters interest in the probability distribution of X, since they express the central tendency and the variability of the v.a. X.

Binomial Distribution

The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail. This is the model suitable for sampling from one infinite population, where p is the fraction of defective or non-compliant elements present in the population.

The random variable X represents the number of “hits” in n independent Bernoulli trials with probability of constant success equal to p in each trial.

Poisson Distribution

Poisson distribution is a statistical distribution that shows how many times an event is likely to occur within a specified period of time. It is used for independent events which occur at a constant rate within a given interval of time. It is the suitable model to model the number of defects or not conformity found in a product unit.

Normal Distribution or Gauss Distribution

The normal distribution is a probability function that describes how the values of a variable are distributed. It has a very recognisable features, in fact most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions. Extreme values in both tails of the distribution are similarly unlikely.