** Probability Distribution Function vs Probability Density Function
**

Probability is the likelihood of an event to happen. This idea is very common, and used frequently in the day to day life when we assess our opportunities, transaction, and many other things. Extending this simple concept to a larger set of events is a bit more challenging. For example, we cannot easily figure out the chances of winning a lottery, but it is convenient, rather intuitive, to say that there is a likelihood of one out of six that we are going get number six in a dice thrown.

When the number of events that can take place is becoming larger, or the number of individual possibilities is large, this rather simple idea of probability fails. Therefore, it has to be given a solid mathematical definition before approaching problems with higher complexity.

When the number of events that can take place in a single situation is large, it is impossible to consider each event individually as like in the example of the dice thrown. Hence, the whole set of events is summarized by introducing the concept of the random variable. It is a variable, which can assume the values of different events in that particular situation (or the sample space). It gives a mathematical sense to simple events in the situation, and mathematical way of addressing the event. More precisely, a random variable is a real value function over the elements of the sample space. The random variables can either be discrete or continuous. They are usually denoted by the uppercase letters of the English alphabet.

Probability distribution function (or simply, the probability distribution) is a function that assigns the probability values for each event; i.e. it provides a relation to the probabilities for the values that the random variable can take. The probability distribution function is defined for discrete random variables.

Probability density function is the equivalent of the probability distribution function for the continuous random variables, gives the likelihood of a certain random variable to assume a certain value.

If *X* is a discrete random variable, the function given as *f*(*x*) = *P*(*X *= *x*) for each *x* within the range of *X* is called the probability distribution function. A function can serve as the probability distribution function if and only if the function satisfies the following conditions.

1. *f*(*x*) ≥ 0

2. ∑ *f*(*x*) = 1

A function *f*(*x*) that is defined over the set of real numbers is called the probability density function of the continuous random variable *X*, if and only if,

*P*(*a *≤ *x *≤ *b*) = _{a}∫^{b }*f*(*x*) *dx* for any real constants *a* and *b*.

The probability density function should satisfy the following conditions too.

1. *f*(*x*) ≥ 0 for all *x*: -∞ < *x* < +∞

2. _{-∞}∫^{+∞ }*f*(*x*) *dx *= 1

Both probability distribution function and the probability density function are used to represent the distribution of probabilities over the sample space. Commonly, these are called probability distributions.

For statistical modeling, standard probability density functions and probability distribution functions are derived. The normal distribution and the standard normal distribution are examples of the continuous probability distributions. Binomial distribution and Poisson distribution are examples of discrete probability distributions.

**What is the difference between Probability Distribution and Probability Density Function?**

• Probability distribution function and probability density function are functions defined over the sample space, to assign the relevant probability value to each element.

• Probability distribution functions are defined for the discrete random variables while probability density functions are defined for the continuous random variables.

• Distribution of probability values (i.e. probability distributions) are best portrayed by the probability density function and the probability distribution function.

• The probability distribution function can be represented as values in a table, but that is not possible for the probability density function because the variable is continuous.

• When plotted, the probability distribution function gives a bar plot while the probability density function gives a curve.

• The height/length of the bars of the probability distribution function must add to 1 while the area under the curve of the probability density function must add to 1.

• In both cases, all the values of the function must be non-negative.

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