6.3: Probability Distributions

The probability of a random variable x is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.

A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson probability distribution.

The binomial distribution is a probability distribution of a procedure with a fixed number of trials, where each trial has only two possible outcomes. A distribution involving coin tossing is an example of this distribution, as a coin toss has only two possible outcomes– heads or tails.

Poisson distribution is a distribution of independent events occurring over a specific interval. The number of messages received per day is an example of this type of distribution. A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial if the probability of success is "small" (less than or equal to 0.05) and the number of trials is "large" (greater than or equal to 20).

Continuous probability distributions are the distributions associated with continuous random variables. They are divided into two categories– uniform distribution and normal distribution,

A uniform distribution is rectangular-shaped, indicating that the values are evenly spread over the range of possibilities. An example would be a distribution of hearts, spades, clubs, and diamonds in a deck of cards. This is because there is an equal probability of drawing a heart, a spade, a club, or a diamond from the card deck.

In contrast, a normal distribution is a probability distribution that forms a symmetric bell-shaped curve. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is extremely important, but it can only be applied to some things in the real world.

This text is adapted from Openstax, Introductory Statistics, Section 4.

A probability distribution is a representation of the probabilities associated with random variables.

It is commonly expressed in the form of a formula, graph, or table.

For any probability distribution, each individual probability needs to be between zero and one, and the sum of the individual probabilities must be equal to one.

Probability distributions can be broadly classified as either discrete or continuous.

Discrete distributions are subdivided into binomial and Poisson distributions.

A binomial distribution describes cases having multiple trials but only two possible outcomes per trial, like the tossing of a coin.

In a Poisson distribution, there can be independent events occurring in specific intervals, such as the number of website visitors per hour.

Similarly, a continuous probability is subdivided into uniform and normal distributions.

The uniform distribution represents probabilities that are evenly spread over the possible range, like the voltage provided by the electric company.

The normal distribution represents probabilities that form a symmetrical bell-shaped graph, for example, the birth weight of babies.

Overall, the probability distributions are useful in estimating the chance of an event occurring.