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:
 

1. A PDF always has a positive value and can never be negative, because probabilities can never be negative.
    
2. 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%.
    
3. 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.
    
4. 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:

1. The PDF must be greater than or equal to zero for all the values of the random variable.
    
2. In a probability density function, the integral of the function must equal one.

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) = ddx[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 ≤ x ≤ b) = F(b) - F(a) = ab f(x)dx

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

P (a ≤ x ≤ 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) =μ =-∞∞xf(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: 

                        -∞M​f(x)dx=M∞f(x)dx=12​

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:
 

1. 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.
    
2. 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.
    
3. 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.