Probability Distribution:

Probability distribution is a measure of random variable’s behavior/dispersion which indicates the likelihood of an event or outcome. To represent a probability distribution, we use equations and tables of variable values and probabilities.

To describe probabilities, statisticians use this notation p(x) = the likelihood that random variable which takes a specific value of x.

Based on the type of Random variable, we have two types of distributions:

• For discrete variables- Discrete probability distributions
• For continuous variables- Probability density functions

Discrete Probability Distributions:

Discrete probability functions / probability mass functions (PMF) can assume a discrete number of values. For example, number of coin tosses and counts of events are discrete functions because there are no in-between values. For example, only heads or tails occur in a single coin toss.

Each possible value has non-zero likelihood. Since total probability must be 1, one of the values must occur for each opportunity.

To display discrete distribution having a finite number of values, we can arrange them in a tabular manner with their corresponding probabilities.

Types of Discrete Probability Distribution:

• Binomial distribution is used for experiments whose outcome is in form of binary data, such as coin tosses.
• Poisson distribution is used for quantitative random variables (countable data), such as the count of check-ins per hour at an airport.
• Uniform distribution is used for multiple events having the same probability, such as rolling a die.

The mathematical representation of a discrete probability function, p(x), is a function that satisfies the following properties:

The probability that x can have a specific value is p(x)

P[X = x] = p(x) = p_x , where X is random variable.

p(x) is non-negative for all Real x i.e., where 0 <= p(x) <= 1

The sum of p(x) over all possible values of x is 1

Where, j represents all possible values that x can have and p_j is the probability at x_j

Continuous Probability Distributions:

Continuous probability functions / probability density functions (PDF) can assume an infinite number of values between any two values such as height, weight, and temperature.

In discrete probability distributions, each particular value has non-zero likelihood (probability), but in continuous distributions, specific values may have a zero probability.

Example:

The likelihood of measuring a temperature that is exactly 32 degrees is zero. Statisticians say that any individual value has an infinitesimally small probability that is equivalent to zero.

Probabilities for continuous distributions are measured over ranges of values rather than single points. Here we calculate probability whether the likelihood of a value falls within an interval or not.

In discrete distributions, the sum of all probabilities must equal one. Similarly, in continuous distributions, the entire area in a probability plot under the distribution curve must be equal to 1. The proportion of the area under the curve that falls within a range of values along the X-axis represents the probability.

Each probability distribution has parameters that define its shape. Most distributions have between 1-3 parameters. Based on these parameters, the shape of the distribution and all of its probabilities can be established, such as the central tendency and the variability.

The mathematical representation of a continuous probability function, f(x)

The probability that x is between two points a and b is

p[a ≤ x ≤ b]=

It is non-negative for all real x.