Discree Probability Model

The discrete probability model is used to describe the probability of each possible outcome when the number of outcomes is finite or countably infinite.
A random variable is a function that assigns a numerical value to each outcome of a statistical experiment.
A discrete random variable can only take a countable number of values. Examples include the number of heads in a coin toss, the number of girls in a family of three children etc.

The probability distribution of a discrete random variable is defined by a probability mass function, denoted by the pmf.
The pmf gives the probability that a discrete random variable is exactly equal to some value. It satisfies two conditions: each probability is between 0 and 1 inclusive, and the sum of the probabilities is 1.
The expected value or mean of a discrete random variable X, denoted E(X), is the average value of X in a large number of experiments.
Similarly, the variance denoted Var(X), measures the dispersion of the random variable around its mean.

Binomial distribution model describes number of successes in a fixed number of Bernoulli trials with the same probability of success.
Poison distribution model describes number of events in a fixed interval of time or space with a known average rate of occurrence.
Geometric distribution model describes number of failures before the first success in a series of independent and identically distributed Bernoulli trials.

The discrete probability model is used to describe the probability of outcomes for discrete random variables.
It’s important to correctly identify and use the appropriate probability model depending on the nature of experiment or event being studied.
Understanding the properties such as pmf, mean, and variance is key to modelling and analysing the probability of occurrence of events.
Mastery of different types of discrete probability models like binomial, Poisson, and geometric distribution models opens up various analytical approaches to solve real-life problems.