Probability distributions are a fundamental concept in statistics. They are used both on a theoretical level and a practical level. To understand probability distributions, it is important to understand variables. Random Variables, and some notation.
A variable is a symbol (A, B, x, y, etc.) that can take on any of a specified set of values.
When the value of a variable is the outcome of a statistical experiment, that variable is a random variable. We normally use a capital letter to represent a random variable and a lower-case letter, to represent one of its values.
Types of Random Variable:
Discrete Random Variables are random variables that has countable values, such as a list of non-negative integers. Discrete random variables can take on a countable number of possible values. This might be a finite set of possible values and could also be a countably infinite set of values. Eg. The number of females in a family, number of heads in 10 toss of a coin, etc.
Continuous Random Variables can take on an infinite number of possible values, corresponding to all values in an interval. The time taken for something to be done is an example of Continuous Random Variables. Eg. Height, Weight, etc.
P(X = x) refers to the probability that the random variable X is equal to a particular value, denoted by x. As an example, P(X = 1) refers to the probability that the random variable X is equal to 1.
Probability distributions are described in many ways in many books and blogs but they mean the same thing. Some of the description are:
1. A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
2. The mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.
3. A table or an equation that links each outcome of a statistical experiment with its probability of occurrence.
Types of Probability Distribution
1. Discrete Probability Distribution
A discrete distribution describes the probability of occurrence of each value of a discrete random variable. The probability distribution of a discrete random variable X is a listing of all possible values of X and their probabilities of occurring. This can be illustrated using a table, histogram, or formula. The probability that x can take a specific value is p(x). That is: P[X=x]=p(x). p(x) is non-negative for all real x. Discrete probability functions are referred to as probability mass functions.
To be a valid discrete probability distribution, two conditions must be satisfied:
1. All probabilities must lie between 0 and 1: 0 ≤ p(x) ≤ 1 for all x.
2. The probabilities must sum to 1 which is ∑ p(x) = 1.
A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a dice).
Types of Discrete Probability Distribution:
i. Bernoulli Distribution
ii. Poisson Distribution
iii. Binomial Distribution
iv. Geometric Distribution
v. Negative Binomial Distribution
vi. Hypergeometric Distribution
2. Continuous Probability Distribution
A continuous probability distribution describes the relative likelihood of all of the possible values of x. It come in a variety of shapes. Probabilities of continuous random variables (X) are defined as the area under the curve of its probability distribution function. The probability that a continuous random variable equals some value is always zero. Continuous probability functions are referred to as probability density functions. Continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day.
The probability that x is between two points a and b is:
It is non-negative for all real x.
The integral of the probability function is one. Mathematically,
Types of continuous probability distribution:
i. Normal Distribution
ii. Uniform Distribution
iii. Chi- Squared Distribution
iv. Beta Distribution
For all the types of continuous as well as discrete distribution, Expected Value (or theoretical mean or expectation) and variance can be calculated using their properties and probability function. I will be doing this in next blog with detail mathematical explanation.
Quiz:
Identify the type of random variablein following cases.
- the number of tosses until heads first appears
2. volume of water in a randomly selected container
3. time it takes to get to college
4. sum of numbers appeared on toss of two fair dices
Reference
1. Jeremy Balka’s YouTube Channel