Probability Density Function

The probability density function (PDF) in statistics describes how likely different outcomes are within a given range. We get the PDF from the cumulative distribution function (CDF) when it is differentiated. Both the cumulative distribution function and the probability density function (PDF) are used to describe the probability distribution of a continuous random variable.

In PDF, the percentage of the dataset’s distribution falling between two criteria is the probability. It is commonly used by financial analysts to know how returns are distributed so they can evaluate the risk and any expectation of investment prices and returns.

Some key takeaways of probability density function are:
 

1. To measure the likelihood that an investment will have returns that fall within a range of values, we use the probability density function.
    
2. PDFs are also used to tell whether a certain investment would be risky or not.
    
3. We usually plot PDFs on graphs, and it typically resembles a bell curve. With the data lying below the curve.

Here are a few properties of probability density function that need to be kept in mind when learning about PDFs:

• A PDF always has a positive value and can never be negative, because probabilities can never be negative.
  
   
• When calculating PDF on a graph, the total area under the curve must be equal to 1. This makes sure that the variable taking any possible value is 100%.
  
   
• A PDF can take values greater than 1 as long as the area under the curve is 1. This can be achieved if the distribution is extremely concentrated in a small range.
  
   
• A probability density function can have one or more peaks, it can even be flat without any distinct peaks, depending on the distribution of data.

There are a few conditions that must be met when calculating the probability density function:

• The PDF must be non-negative for all values of the random variable.
  This means that the function f(x) can never be negative. In probability, negative values don’t make sense; you can’t have a “negative chance” of something happening. So, for every possible value of the random variable X, the function must satisfy:
  
  \(f(x) \geq 0 \)
  
  This ensures the PDF correctly represents real, valid probabilities.
  
   
• The integral of the PDF over its entire range must equal 1.
  A probability density function describes how probability is spread across all possible values of a continuous random variable. Since the total probability of all outcomes must be 1 (or 100%), the area under the entire PDF must also be 1.
  
  \(\int_{-\infty}^{\infty} f(x)\, dx = 1 \)
  
  This condition guarantees that the function represents a complete and valid probability distribution. Without this, you wouldn’t know where all the probability “went.”

There are several important types of probability density functions (PDFs). The most common ones include:

• Uniform distribution
• Binomial distribution
• Normal distribution
• Chi-square distribution

Uniform distribution:
A uniform distribution represents outcomes that are equally likely within a specific interval.
It is also known as the rectangular distribution because its graph forms a rectangle. It is also known as the rectangular distribution because its graph forms a rectangle. 

This distribution is written as U(a, b), where:
a = minimum value
b = maximum value
PDF formula for uniform distribution
For any value x between a and b:
 

\(f(x) = \frac{1}{b - a} \)

Binomial distribution:
The binomial distribution describes the probability of getting a certain number of successes in a fixed number of trials. It has two parameters.

• n = number of trials
• P = probability of success
   

PDF formula for binomial distribution
If:
 

• x = number of successes
• q = 1 - p = probability of failure
   

Then,

\(P(x) = \binom{n}{x} p^x q^{\,n-x} \)

Normal distribution:
The normal distribution, also called as the Gaussian distribution, is symmetric and shaped like a bell curve. It is written as N(μ, σ2), where:
 

• μ = mean
• \(σ^2\) = variance
   

PDF formula for nominal distribution
 

\(f(x)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \)
 

Standard normal distribution
When,
Mean μ = 0
Standard deviation σ = 1
The formula becomes:
 

\(f(x)=\frac{1}{2\pi} e^{-\frac{x^2}{2}} \)

Chi - square distribution:
A chi-square distribution represents the sum of squares of k independent standard normal variables. It is written as \(X^2(k)\), where k is the degrees of freedom.
PDF formula for chi - square distribution
 

For \(x>0\),

\(f(x) = \frac{1}{2^{k/2}\,\Gamma(k/2)} \, x^{(k/2)-1} e^{-x/2}, \quad x > 0 f(x) = 0, \quad x \le 0 \)

The main differences between a probability density function (PDF) and a cumulative distribution function (CDF) are summarized in the table below:

To calculate the probability density function of a continuous random variable, we use the following formula:

Let us take ‘x’ as the continuous random variable and F(x) as the cumulative distribution function (CDF) of x. So the formula for the PDF, f(x) will be:

\(f(x) = \frac{d}{dx}[F(x)] = F'(x) \)

Now, let us say you want to find the probability that X lies between a and b. Then the formula to be used is:

\(P(a \le x \le b) = F(b) - F(a) = \int_a^b f(x) \, dx \)

Another way to express the probability would be by using the CDF, which is:

\(P(a \le x \le b) = F(b) - F(a) \)

The probability density function is represented graphically by plotting the function f(x) against the values of the random variable x. The below graph shows the probability of X being between two points a and b.

The graph usually looks like a bell curve, and the probability for the random variable is the area under the curve. This area must be equal to 1. 

Mean of Probability Density Function
The mean of PDF refers to the average value of the random variable. We call this mean the expected value, and we denote it as μ or E(X), where X is the random variable. We express the mean of the probability density function f(x) for the random variable x as:

\(E(X) = \mu = \int_{-\infty}^{\infty} x f(x) \, dx \)

Median of Probability Density Function
The value dividing the PDF graph into two halves is the median. If x = M is the median, the area under the curve from -∞ to M and ∞ to M are equal, then the median value = ½. The median of the probability density function is expressed as: 

\(\int_{-\infty}^{M} f(x) \, dx = \int_{M}^{\infty} f(x) \, dx = \frac{1}{2} \)

The probability density function is a complex mathematical topic, and to better understand it, the following tips and tricks are provided.

 

• Remember that a PDF describes how probabilities are distributed over continuous values; it gives the likelihood density, not the probability itself.
  
   
• The Cumulative Distribution Function (CDF) gives the probability that a random variable is less than or equal to a specific value. The CDF is obtained by integrating the PDF.
  
   
• Draw or visualize the shape of common PDFs like Normal, Exponential, Uniform, and Beta distributions. This helps you quickly recognize their properties during problem-solving.
  
   
• Since probabilities are calculated by integrating the PDF, strengthen your integration techniques, especially for exponential and polynomial functions.
  
   
• Relate PDFs to real data, like height distribution, rainfall measurement, or machine failure rates. Real examples make concepts easier to grasp and remember.
  
   
• Parents and teachers can encourage drawing and visualizing graphs of common PDFs, such as the usual, exponential, uniform, and beta distributions.
  
   
• Teachers can use step-by-step examples, while children can start with simple functions to build confidence.
  
   
• Children can work together on real examples to make the topic more relatable and memorable.

Probability density functions are used in various fields where they use it to model and analyze data. Here are some of the real-world applications:
 

• Finance and Risk Management: PDFs are used to model the distribution of stock prices. In risk management, PDFs help in calculating the value at risk (VAR) to estimate any kind of potential loss.

• Healthcare: When we want to know the patient's blood pressure, cholesterol levels, or survival time, PDFs are used to model the distribution. This helps in understanding the spread of any diseases or the effectiveness of the treatments.
  
   

• Environmental Science: PDFs are used to predict any kind of extreme weather events, or track the air pollution in an area or country, rainfall, and temperature.
  
   
• Engineering and Reliability Analysis: PDFs are used to model the lifespan of components or systems (like engines, circuits, or machines). Engineers use these models to estimate the probability of failure over time and improve product reliability.
  
   
• ​​​​​​​Telecommunications: In signal processing, PDFs help describe noise distributions in communication channels. This allows engineers to design systems that minimize transmission errors and improve data accuracy.