Probability Density Function

A probability density function is a function that provides the likelihood that the value of a random variable will fall within a specific range of values. This likelihood is provided by the function itself. In the case of continuous random variables, we make use of a function known as the probability density function. We use a function called the probability mass function, which is very similar to the probability density function, when dealing with discrete random variables.

The graph of a probability density function looks like a bell curve because it represents the distribution of possible outcomes. The probability of the outcome of the designated observation can be calculated using the region of space that exists between any two specified values. The probabilities associated with a continuous random variable can be determined by successfully completing the integral of this function. Within the scope of this article, we will perform an in-depth analysis of the probability density function and investigate the myriad of facets that are associated with it.

Probability density function

It is possible to define the distribution of continuous random variables by utilising the probability density function as well as the cumulative distribution function. The probability density function is the result that we get when we differentiate the cumulative distribution function of a continuous random variable. On the other hand, the cumulative distribution function is what we obtain when we integrate the probability density function.

The Definition of the Probability Density Function

The density of the probability that a continuous random variable will lie within a particular range of values is what the probability density function is attempting to define. We integrate the probability density function between the two points that you have specified so that we can determine this probability.

Illustration of a Probability Density Function

Imagine that we have a continuous random variable, and that the probability density function for this variable is given by the equation f(x) = x + 2, for all values of x between 0 and 2. It is important that we determine P(0.5 < X < 1).The next step is to integrate x +2 while observing the boundaries of 0.5 and 1. This gives us 1.375 as our answer. As a result, the probability that the continuous random variable is somewhere between 0.5 and 1 is equal to 1.375.

The Formula for the Probability Density Function

A continuous random variable has a function that is analogous to the probability mass function of a discrete random variable. This function is called the probability density function. The evaluation of discrete random variables can take place at a specific point, whereas the evaluation of continuous random variables must take place within a predetermined time interval. This is due to the fact that there is no chance that a continuous random variable will end up with an exact value. The various formulas for the probability density function are presented in the following list.

Probability Density Function Continuous Random Variable

Let us assume that the variable X is a continuous random variable. Consider the cumulative distribution function of X to be denoted by F(x). The formula for the probability density function, which is denoted by f(x), can then be expressed as follows:

f(x) = d F( x )/dx =F’ (x)

Using the probability density function, we can express the probability that X is located between the lower limit ‘a’ and the upper limit ‘b’ in the following way:

P(a < X ≤ b) = F(b) – F(a) = ^b∫_a b a f ( x ) d x

In this context, the cumulative distribution function at b is denoted by F(b), and the cumulative distribution function at an is denoted by F(a).

A Graph Showing the Probability Density Function

The probability density function will give the probability distribution of the continuous random variable X if it is assumed that X is a random variable that changes over time. The probability that X will be located between the two points a and b is illustrated in the graph that can be found below.

Properties of Probability Density Function

The properties of the probability density function assist in the resolution of problems in a more expedient manner. In the event that f(x) represents the probability distribution of a continuous random variable, X, then the following are some of the advantageous properties that it possesses:

•f(x) ≥ 0. This suggests that the probability density function for any real number can either be greater than 0 or equal to 0. 

•However, it is not possible for it to ever be less than 0 or in a negative value. 

^∞∫_-∞ f ( x ) d x = 1. Therefore, there will be an equal amount of space below the entire probability density curve.

Conclusion

A probability density function is a function that provides the likelihood that the value of a random variable will fall within a specific range of values.It is possible to define the distribution of continuous random variables by utilising the probability density function as well as the cumulative distribution function. The probability density function is the result that we get when we differentiate the cumulative distribution function of a continuous random variable.A continuous random variable has a function that is analogous to the probability mass function of a discrete random variable. This function is called the probability density function.The properties of the probability density function assist in the resolution of problems in a more expedient manner. In the event that f(x) represents the probability distribution of a continuous random variable, X.