Probability Density Function : Meaning, Formula, and Graph

What is the Probability Density Function?

Probability Density Function (PDF) and Cumulative Distribution Function (CDF) describe the probability distribution of a continuous random variable. In simpler terms, PDF tells about how likely different values of the continuous random variable are. By differentiating the CDF of a continuous random variable, we can determine the Probability Density Functions. In the same way, if we integrate the Probability Density Function, we can obtain the probability or CDF between two specific points.

For example, Suppose there is a continuous random variable with PDF given by f(x) = x + 4, where 0 < x ≤ 4; and we want to determine P(1.5 < X < 2). To find the probability of the given continuous random variable, we will integrate x + 4 within the limits of 1.5 and 2. The result will be 2.875. Hence, the probability of X ranging between 1.5 and 2 is 2.875.

Probability Density Function Formula

Function Formula in case of Discrete Random Variables

• The PDF of discrete random variables gives the probability of the variable taking a specific value.
• The PDF formula is denoted as P(X=x), where X is the random variable, and x is a particular value it can take.
• A function can be quantified as a Probability Function or Probability Density Function of a discrete random variable, when

f_X(x) ≥ 0, for all x within the range of X

Example:

Suppose there are 6 balls in a bag. The random variable X is the weight of a ball (in kg) selected at random. Balls 1, 2, and 3 weighs 0.5 kg; Balls 4 and 5 weighs 0.25 kg; and Ball 6 weighs 0.3 kg. Write the Probability Density Function for X.

Solution:

f_X(0.5) = P(X = 0.5) = 

f_X(0.25) = P(X = 0.25) = 

f_X(0.3) = P(X = 0.3) = 

Function Formula in case of Continuous Random Variables

• The PDF of continuous random variables gives the likelihood of the variable falling within a specific interval.
• The PDF is denoted as f(x), where x is the continuous random variable.
• The probability of x lying within a given interval [a,b] is calculated by integrating the PDF over that range. Therefore,

• A function will serve as a PDF when it follows the following conditions:

f_X(x) ≥ 0, -∞ ≤ x ≤ ∞

Here the limits are taken as -∞, ∞. However, in real life, we will use the values of x, applicable to the question. It means that if the variable is given between the range of a and b (a < x < b), then the limits for integration will be a and b.

Example:

A continuous random variable Y has PDF f_Y(y) = 10y²(1-y), where 0 < y < 2. Determine P(Y < 0.2).

Solution:

P(Y < 0.2) = 

P(Y < 0.2) = 

P(Y < 0.2) = 0.023

Probability Density Function Graph

The probability density function (PDF) is found by adding up the density of the variable over a certain range. We use the symbol f(x) to represent this function. At any point on the graph, the value of this function is positive or zero. When we calculate the definite integral of the PDF over the entire range, the result is always one.

The graph of PDFs often looks like a bell curve, with the likelihood of outcomes shown below the curve. The figure below illustrates the graph of a probability density function for a continuous random variable x, with the function represented by f(x).

Properties of Probability Density Function

1. The density function, denoted as f(x), is used for a continuous random variable with values between specific limits, a and b. To find the Probability Density Function (PDF), we calculate the area under the curve between these limits on the X-axis.

2. The PDF, f(x), is always greater than or equal to zero for any possible value of x.

3. The total area under the density curve and the X-axis, within the given range from a to b, is equal to 1. This represents the entire probability space for the continuous random variable.

4. The density function curve extends smoothly over the entire specified range, illustrating the continuous nature of the random variable.

5. The PDF is defined across a range of continuous values, reflecting the variable’s domain.

Probability Distribution Function of Discrete Distribution

I. Discrete Uniform Distribution

The Discrete Uniform Distribution represents outcomes where each possible value has an equal chance of occurring. For example, rolling a fair six-sided die is a classic case of discrete uniform distribution.

II. Bernouilli Distribution

The Bernoulli Distribution is applicable to situations with two possible outcomes, typically labeled as success and failure. It is often used in scenarios like coin flips, where success might be getting heads.

P(X = x) = p^x (1-p)^1-x, x = 0, 1; 0 < p < 1

III. Binomial Distribution

Binomial Distribution deals with the number of successes in a fixed number of independent Bernoulli trials. For instance, determining the probability of getting a certain number of heads in multiple coin flips.

P(X = x) = ⁿC_x p^x (q)^n-x, x = 0, 1, 2,……,n; 0 < p < 1

IV. Geometric Distribution

Geometric Distribution models the number of trials needed for the first success in a sequence of independent Bernoulli trials. For example, finding the probability of the first successful free throw in basketball.

P(X = x) = (1 – p)^k-1.p

V. Negative Binomial Distribution

The Negative Binomial Distribution focuses on the number of trials needed for a fixed number of successes in a sequence of independent Bernoulli trials. It’s applicable to scenarios like predicting the number of attempts to make three successful shots in basketball.

VI. Poisson Distribution

The Poisson Distribution is useful for events with a known average rate of occurrence within a fixed interval. It’s commonly employed in areas, such as predicting the number of emails received in an hour.

Probability Distribution Function of Continuous Distribution

I. Continuous Uniform Distribution

The Continuous Uniform Distribution represents a constant probability for all values within a specified range. The Probability Distribution Function (PDF) for this distribution is a flat, horizontal line, indicating equal likelihood for any value within the given interval.

where a and b are the lower and upper limits of the distribution.

II. Gamma Distribution

The Gamma Distribution is used to model the time until a certain number of events occur in a Poisson process. Its PDF is characterized by a shape parameter and a rate parameter, providing flexibility in describing various scenarios, including waiting times and reliability analysis.

III. Exponential Distribution

The Exponential Distribution is often employed to model the time between events in a Poisson process. Its PDF exhibits a decreasing exponential curve, emphasizing a constant hazard rate and frequent use in reliability and queuing studies.

IV. Chi-Square Distribution

The Chi-Square Distribution is widely utilized in statistical hypothesis testing. Its PDF is determined by the degrees of freedom, influencing the shape of the distribution. Common applications include testing variance and goodness of fit.

V. Beta Distribution

The Beta Distribution is versatile, representing random variables bounded within a specific interval. Its PDF is defined by shape parameters, allowing it to model a range of outcomes, including proportions and probabilities.

VI. Lognormal Distribution

The Lognormal Distribution is suitable for variables that follow a log-normal pattern. Its PDF is characterized by parameters influencing the shape and scale of the distribution. Applications include modelling stock prices and certain biological processes.

VII. Normal Distribution

The Normal Distribution, or Gaussian Distribution, is a fundamental distribution found in many natural phenomena. Its PDF is the famous bell-shaped curve, determined by mean and standard deviation parameters. It is extensively used in statistical analyses due to the Central Limit Theorem and its prevalence in various fields.

Applications of Probability Density Function

Probability Density Function can be used in various fields and problems:

• Probability Assessment: It helps in assessing the probability of a continuous random variable falling within a specific range of values.
• Statistical Analysis: The PDF is a crucial tool in statistical analysis, providing insights into the distribution of continuous data.
• Area under the Curve: The area under the PDF curve between two points represents the probability of the variable lying within that range.
• Comparisons: It enables comparisons between different continuous random variables that help in understanding their respective likelihoods.
• Research and Sciences: PDF is widely used in fields like physics, engineering, and finance for modelling and analyzing continuous phenomena.
• Integration in Calculations: The PDF is often used in calculus, where integrating it over a range yields the probability of the random variable falling within that interval.
• Probability Density Assessment: It allows for the assessment of how densely the probability is distributed across different values of a continuous random variable.
• Understanding Distribution Shapes: The shape of the PDF curve provides insights into the characteristics of the underlying probability distribution.
• Risk Analysis: In finance and insurance, PDF is used to assess the probability of different financial outcomes and potential risks.
• Mathematical Basis: It serves as a fundamental concept in probability theory, providing a mathematical foundation for understanding continuous random variables.