Probability Distribution

The probability distribution function is also known as the cumulative distribution function (CDF). If there is a random variable, X, and its value is evaluated at a point, x, then the probability distribution function gives the probability that X will take a value lesser than or equal to x. It can be written as F(x) = P (X ≤ x). Furthermore, if there is a semi-closed interval given by (a, b] then the probability distribution function is given by the formula P(a < X ≤ b) = F(b) - F(a). The probability distribution function of a random variable always lies between 0 and 1. It is a non-decreasing function.

A probability distribution of a random variable can be described by a probability distribution function (CDF) and a probability mass function (for discrete random variables) or a probability density function (for continuous random variables). Depending upon the type of distribution a random variable follows there can be different formulas for a probability distribution.

Probability Distribution of a Random Variable

A random variable can be described as a variable that can take on the possible values of an outcome of an experiment. There can be two types of random variables, namely, discrete and continuous random variables. Given below are the various formulas for the probability distribution of a random variable.

Probability Distribution of a Discrete Random Variable

A discrete random variable can be defined as a variable that can take a countable distinct value like 0, 1, 2, 3... The formulas for the probability distribution function and the probability mass function for a discrete random variable are given below:

• Probability Distribution Function: F(x) = P (X ≤ x)
• Probability Mass Function: p(x) = P(X = x)

Probability Distribution of a Continuous Random Variable

A continuous random variable can be defined as a variable that can take on infinitely many values. As the probability that a continuous random variable will take on an exact value is 0 hence, we cannot use the probability mass function (pmf) to describe such a distribution. We use the probability density function in place of the pmf. The formulas for the probability distribution of a continuous random variable are given below:

• Probability Distribution Function: F(x) = P (X ≤ x)
• Probability Density Function: f(x) = d/dx (F(x)) where F(x) = \(\int_{-\infty }^{x}f(u)du\).

Probability Distribution of a Normal Distribution

A normal distribution is a type of continuous probability distribution. The mean and the variance are the two parameters required to describe such a distribution. If X is a random variable that follows a normal distribution then it is denoted as \(X\sim N(\mu,\sigma ^{2})\). The probability distribution formulas are given below:

• Probability Distribution Function: F(x) = \(P = \left ( Z \leq \frac{x - \mu }{\sigma } \right ) = \Phi \left ( \frac{x - \mu }{\sigma } \right )\)
• Probability Density Function: f(x) = \(\frac{1}{\sigma \sqrt{2\Pi }}e^{-\frac{(x-\mu)^{2} }{2\sigma ^{2}}}\)

Probability Distribution of a Geometric Distribution

A geometric distribution is a type of discrete probability distribution where the random variable, X, represents the number of Bernoulli trials required till the first success is obtained. The outcome of each trial can either be a success (p) or a failure (1 - p). Given below are the formulas for the probability distribution of a geometric distribution.

• Probability Distribution Function: F(x) = P (X ≤ x) = 1 - (1 - p)^x
• Probability Mass Function: P(X = x) = (1 - p)^{x - 1}p

Probability Distribution of a Binomial Distribution

A binomial distribution is another type of discrete probability distribution that gives the number of successes when a sequence of n independent experiments is conducted. The outcome of each experiment can be either a success or a failure. It is denoted as \(X\sim Bin(n,p)\)The formulas for the probability distribution of a binomial distribution are given below:

• Probability Distribution Function: F(x) = P (X ≤ x) = \(\sum_{i = 0}^{x}\binom{n}{i}p^{i}(1 - p)^{n - i}\)
• Probability Mass Function: P(X = x) = \(\binom{n}{x}p^{x}(1 - p)^{n - x}\)

A probability distribution graph helps to give a visual approach of the distribution that a given random variable follows. For continuous distributions, the area under a probability distribution curve must always be equal to one. This is analogous to discrete distributions where the sum of all probabilities must be equal to 1.

For a continuous random variable, X, the probability density function is used to obtain the probability distribution graph. Suppose X lies between a and b, then the probability distribution graph is given as follows:

If a discrete random variable follows a probability distribution such as a Bernoulli distribution, then the probability distribution graph is given as follows:

The outcome of a Bernoulli trial can either be 0 or 1. Thus, the random variable, X, can only take on the values 0 or 1 as depicted in the graph.

Both the probability distribution function and the probability density function are used to describe a probability distribution. A probability distribution function is used to summarize the probability distribution of a random variable. Such a function is well-defined for both continuous and discrete probability distributions. A probability density function (pdf), on the other hand, can only be used for continuous distributions. It can be defined as the likelihood that a continuous random variable, X, will take on a value that lies between a given range of values. For discrete distributions, a probability mass function (pmf) is used which is analogous to the probability density function. This function gives the probability that a random variable will exactly take on a specific value. We use the pdf in place of the pmf for continuous distributions as the probability that a continuous random variable will take on an exact value is 0.

Related Articles:

• Probability and Statistics
• Experimental probability
• Probability Rules

Important Notes on Probability Distribution

• A probability distribution is used to describe all the possible values of a random variable and their corresponding occurrence probabilities.
• There can be two types of probability distributions. These are the continuous probability distribution (e.g., Normal distribution) and the discrete probability distribution (e.g., Bernoulli distribution).
• A probability distribution function and a probability density function (pdf) can be used to describe the characteristics of a continuous distribution.
• A discrete distribution can be defined by a probability mass function (pmf) and probability distribution function.