Probability Functions

The probability function is a mathematical formula that describes how likely it is for a random variable to fall within a range of particular values rather than an exact value. This function can be used to determine the behaviour of continuous probability systems. Most probability function graphs resemble a bell curve that shows the chances of favourable results under the curve. We will learn about different probability distributions. Let us learn in detail what probability function is and what are its properties.

What is the probability function?

Generally, probability function refers to the concentration of the continuous random variables falling within a  range of specified values. Therefore, the probability function provides a probability of values for continuous random variables. To calculate the probability function of a continuous random variable that takes values between certain limits, say x and y, the area under the curve and the X-axis must be calculated within the lower limit (x) and upper limit (y). Thus, the PDF is given by

P(a)= xy∫f(a) da

Properties

• The X-axis is an asymptote to the curve (an asymptote is a line whose distance from the curve approaches zero). 

• There can be no negative values of the probability function, i.e. f(x)>0, for all x.

• In the density plot, the area covered within the X-axis and the density curve equals 1 because we know all the probabilities of an experiment add up to 1. Therefore, 

P(a)=0∫f(a) da=1

• The variable is defined throughout the continuous values.

How to write a probability function?

If there are infinite values a continuous random variable A can take, the probability of getting one outcome f(a) becomes so less that it tends to zero. So we have to take a specific range and calculate the likelihood of A lying within the interval (x, y). So the probability of getting an outcome, P (x < A < y), can be written as,  

P(x Ay)= xy∫ f(a) da

When A is continuous, the endpoints of intervals while finding probabilities of continuous random variables are ignored. Therefore, for any constants x and y,

P(xAy) =P(x <A< y)

Probability Function Curve – Bell Curve

Probabilistic density is defined as a function that integrates density variably over a range. Its notation is f (x). Most probability function graphs resemble a bell curve that shows the chances of favourable results under the curve. The figure below depicts the probability function graph for a continuous random variable x with function f(x). 

Discrete Uniform Distribution (U)

When all possible outcomes have the same probability, it is called a uniform distribution. X, a discrete random variable having a uniform distribution between a and b can be expressed as X ~ U (a, b).

Binomial distribution (B)

A binomial distribution helps in computing the number of successes in an independent discrete probability distribution. The expression is X ~ B (n, p). A binomial trial can have two outcomes: Win/Loss, head/tail, Pass/Fail, etc.

The probability of success is p, and the probability of failure is q = 1-p. Success is denoted by 1, and Failure is denoted by 0.

Bernoulli distribution (Bern)

A Bernoulli distribution can be viewed as a one time trial of a Binomial experiment. It is expressed as X ~ Bern (p).

X ~ Bern (p) —-> X ~ B (1, p)

The Mean and Variance are:

Mean = p

Variance = p (1-p) = p q

Poisson distribution (Po)

Poisson Distributions are discrete probabilistic distribution functions that show probabilities of events occurring within a fixed time interval.

It is expressed as X ~ Po (λ). The Mean and Variance are:

Mean = Variance = λ

Normal or Gaussian distribution (N)

Normal distributions describe the probability distribution of continuous random variables with real values. The Mean and Variance are:

Mean = μ

Variance = σ2

Standard Normal Distribution (SND)

It is expressed as Z ~ N (0, 1).

The Mean and Standard deviation are given as follows:

Mean = 0

Standard Deviation = 1 

The standardisation formula is Z = (X-μ)/σ

Applications

Various fields of study use probability functions, including statistics, research, engineering and science. Listed below are a few of the most significant applications of probability functions:

• This method in statistics calculates a random variable’s probabilities.

• A probability function is employed in modelling the temporal distribution of atmospheric  concentrations.

• The combustion of diesel engines is modelled using this function.

•  A confidence interval is calculated for a parameter, and a critical region is calculated for a hypothesis.

• An appropriate distributional model can often be determined for univariate data.

• A distributional assumption is often used to create statistical intervals and hypothesis tests. The assumption we are making for the given data set must be justified before computing an interval or test based on it. The distribution used here does not need to be the closest fit for the data. However, it should be adequate to allow valid conclusions to be drawn from the statistical analysis.

• There is often a need for simulation studies to be conducted using random numbers computed according to a specific probability distribution.

Importance of probability function

It is possible to use probability distribution functions in several ways, such as quantity experimenting and describing random variables, determining the statistical significance of estimated parameter values, predicting the probability of a given outcome, and calculating the chances of an outcome falling within a given range.

Statistics revolves around probability distributions, a concept used in both theoretical and practical ways.

Conclusion

In this article, we have learned what a probability function is, its importance and its uses. Various fields of study use probability functions, including statistics, research, engineering and science. Most Probability function graphs resemble a bell curve, with the chances of outcomes below the curve. However, there can be no negative values of the probability function. PMF (probability mass function) is for the discrete random variables, while probability function is for the continuous random variables.