Probability Density Function

Probability Density Function is the function of probability defined for various distributions of variables and is the less common topic in the study of probability throughout the academic journey of students. However, this function is very useful in many areas of real life such as predicting rainfall, financial modelling such as the stock market, income disparity in social sciences, etc.

This article explores the topic of the Probability Density Function in detail including its definition, condition for existence of this function, as well as various examples.

What is Probability Density Function(PDF)?

Probability Density Function is used for calculating the probabilities for continuous random variables. When the cumulative distribution function (CDF) is differentiated we get the probability density function (PDF). Both functions are used to represent the probability distribution of a continuous random variable.

The probability density function is defined over a specific range. By differentiating CDF we get PDF and by integrating the probability density function we can get the cumulative density function.

Probability Density Function Definition

Probability density function is the function that represents the density of probability for a continuous random variable over the specified ranges.

Probability Density Function is abbreviated as PDF and for a continuous random variable X, Probability Density Function is denoted by f(x).

PDF of the random variable is obtained by differentiating CDF (Cumulative Distribution Function) of X. The probability density function should be a positive for all possible values of the variable. The total area between the density curve and the x-axis should be equal to 1.

Necessary Conditions for PDF

Let X be the continuous random variable with probability density function f(x). For a function to be valid probability function should satisfy below conditions.

• f(x) ≥ 0, ∀ x ∈ R
• f(x) should be piecewise continuous.
• \int\limits^{\infin}_{-\infin}f(x)dx = 1

So, the PDF should be the non-negative and piecewise continuous function whose total value evaluates to 1.

Check: Normal distribution Formula

Example of a Probability Density Function

Let X be a continuous random variable and the probability density function pdf is given by f(x) = x - 1 , 0 < x ≤ 5. We have to find P (1 < x ≤ 2).

To find the probability P (1 < x ≤ 2) we integrate the pdf f(x) = x - 1 with the limits 1 and 2. This results in the probability P (1 < x ≤ 2) = 0.5

Probability Density Function Formula

Let Y be a continuous random variable and F(y) be the cumulative distribution function (CDF) of Y. Then, the probability density function (PDF) f(y) of Y is obtained by differentiating the CDF of Y.

f(y) = \bold{\frac{d}{dy}[F(y)]} = F'(y)

If we want to calculate the probability for X lying between the interval a and b, then we can use the following formula:

P (a ≤ X ≤ b) = F(b) - F(a) = \bold{\int\limits^{b}_{a}f(x)dx}

Key Points about PDF Formula

• If we differentiate CDF, we get the PDF of the random variable.

f(y) = \bold{\frac{d}{dy}[F(y)]}

• If we integrate PDF, we get the CDF of the random variable.

F(y) = \bold{\int\limits^{y}_{-\infin}f(t) dt}

What Does a Probability Density Function (PDF) Tell Us?

A Probability Density Function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value. Unlike discrete random variables, where probabilities are assigned to specific outcomes, continuous random variables can take on any value within a range. Probability Density Function (PDF) tells us

• Relative Likelihood
• Distribution Shape
• Expected Value and Variance, etc.

How to Find Probability from Probability Density Function

To find the probability from the probability density function we have to follow some steps.

Step 1: First check the PDF is valid or not using the necessary conditions.

Step 2: If the PDF is valid, use the formula and write the required probability and limits.

Step 3: Divide the integration according to the given PDF.

Step 4: Solve all integrations.

Step 5: The resultant value gives the required probability.

Graph for Probability Density Function

If X is continuous random variable and f(x) be the probability density function. The probability for the random variable is given by area under the pdf curve. The graph of PDF looks like bell curve, with the probability of X given by area below the curve. The following graph gives the probability for X lying between interval a and b.

Probability Density Function Properties

Let f(x) be the probability density function for continuous random variable x. Following are some probability density function properties:

• Probability density function is always positive for all the values of x.

f(x) ≥ 0, ∀ x ∈ R

• Total area under probability density curve is equal to 1.

\bold{\int\limits^{\infin}_{-\infin}f(x)dx =1}

• For continuous random variable X, while calculating the random variable probabilities end values of the interval can be ignored i.e., for X lying between interval a and b

P (a ≤ X ≤ b) = P (a ≤ X < b) = P (a < X ≤ b) = P (a < X < b)

• Probability density function of a continuous random variable over a single value is zero.

P(X = a) = P (a ≤ X ≤ a) = \bold{\int\limits^{a}_{a}f(x)dx} = 0

• Probability density function defines itself over the domain of the variable and over the range of the continuous values of the variable.

Mean of Probability Density Function

Mean of the probability density function refers to the average value of the random variable. The mean is also called as expected value or expectation. It is denoted by μ or E[X] where, X is random variable.

Mean of the probability density function f(x) for the continuous random variable X is given by:

\bold{E[X] = \mu = \int\limits^{\infin}_{-\infin}xf(x)dx}

Median of Probability Density Function

Median is the value which divides the probability density function graph into two equal halves. If x = M is the median then, area under curve from -∞ to M and area under curve from M to ∞ are equal which gives the median value = 1/2.

Median of the probability density function f(x) is given by:

\bold{\int\limits^{M}_{-\infin}f(x)dx = \int\limits^{\infin}_{M}f(x)dx=\frac{1}{2}}

Variance Probability Density Function

Variance of probability density function refers to the squared deviation from the mean of a random variable. It is denoted by Var(X) where, X is random variable.

Variance of the probability density function f(x) for continuous random variable X is given by:

Var(X) = E [(X - μ)²] = \bold{\int\limits^{\infin}_{-\infin}(x-\mu)^2f(x)dx}

Standard Deviation of Probability Density Function

Standard Deviation is the square root of the variance. It is denoted by σ and is given by:

σ = √Var(X)

Probability Density Function Vs Cumulative Distribution Function

The key differences between Probability Density Function (PDF) and Cumulative Distribution Function (CDF) are listed in the following table:

Types of Probability Density Function

There are different types of probability density functions given below:

• Uniform Distribution
• Binomial Distribution
• Normal Distribution
• Chi-Square Distribution

Probability Density Function for Uniform Distribution

The uniform distribution is the distribution whose probability for equally likely events lies between a specified range. It is also called as rectangular distribution. The distribution is written as U(a, b) where, a is the minimum value and b is the maximum value.

Probability Density Function for Uniform Distribution Formula

If x is the variable which lies between a and b, then formula of PDF of uniform distribution is given by:

f(x) = 1/ (b - a)

Probability Density Function for Binomial Distribution

The binomial distribution is the distribution which has two parameters: n and p where, n is the total number of trials and p is the probability of success.

Probability Density Function for Binomial Distribution Formula

Let x be the variable, n is the total number of outcomes, p is the probability of success and q be the probability of failure, then probability density function for binomial distribution is given by:

P(x) = ⁿC_x p^x q^n-x

Probability Density Function for Normal Distribution

The normal distribution is distribution that is symmetric about its mean. It is also called as Gaussian distribution. It is denoted as N (\bar{x}, σ²) where, \bar{x}is the mean and σ² is the variance. The graph of the normal distribution is bell like graph.

Probability density function for Normal distribution or Gaussian distribution Formula

If x be the variable, \bar{x} is the mean, σ² is the variance and σ be the standard deviation, then formula for the PDF of Gaussian or normal distribution is given by:

N (\bar{x}, σ²) = f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-1}{2}[\frac{x - \mu}{\sigma}]^2}

In standard normal distribution mean = 0 and standard deviation = 1. So, the formula for the probability density function of the standard normal form is given by:

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

Probability Density Function for Chi-Squared Distribution

Chi-Squared distribution is the distribution defined as the sum of squares of k independent standard normal form. IT is denoted as X²(k).

Probability Density Function for Chi-Squared Distribution Formula

The probability density function for Chi-squared distribution formula is given by:

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

f(x) = 0, otherwise

Joint Probability Density Function

The joint probability density function is the density function that is defined for the probability distribution for two or more random variables. It is denoted as f(x, y) = Probability [(X = x) and (Y = y)] where x and y are the possible values of random variable X and Y. We can get joint PDF by differentiating joint CDF. The joint PDF must be positive and integrate to 1 over the domain.

Difference Between PDF and Joint PDF

The PDF is the function defined for single variable whereas joint PDF is the function defined for two or more than two variables, and other key differences between these both concepts are listed in the following table:

Applications of Probability Density Function

Some of the applications of Probability Density function are:

• Probability density functions are used in statistics for calculating probabilities for random variables.
• It is used in modelling various scientific data.

Read More,

• Cumulative Frequency Distribution
• Probability Distribution Function

Examples on Probability Density Function

Example 1: If the probability density function is given as: \bold{f(x)= \begin{cases} x / 2 & 0\leq x < 4\\ 0 & x\geq4 \end{cases}} . Find P (1 ≤ X ≤ 2).

Solution:

Apply the formula and integrate the PDF.

P (1 ≤ X ≤ 2) = \int\limits^{2}_{1}f(x)dx

f(x) = x / 2 for 0 ≤ x ≤ 4

⇒ P (1 ≤ X ≤ 2) = \int\limits^{2}_{1}(x/2)dx

⇒ P (1 ≤ X ≤ 2) = \frac{1}{2}\times\big [\frac{x^2}{2} \big ]^2_1

⇒ P (1 ≤ X ≤ 2) = 3 / 4

Example 2: If the probability density function is given as: \bold{f(x)= \begin{cases} c(x - 1) & 0 < x < 5\\ 0 & x\geq5 \end{cases}} . Find c.

Solution:

For PDF:

\int\limits^{\infin}_{-\infin}f(x)dx = 1\\ \Rightarrow\int\limits^{1}_{-\infin}f(x)dx\hspace{0.1cm}+\hspace{0.1cm} \int\limits^{5}_{1}f(x)dx \hspace{0.1cm}+\hspace{0.1cm} \int\limits^{\infin}_{5}f(x)dx = 1\\ \Rightarrow\int\limits^{1}_{-\infin}0dx\hspace{0.1cm}+\hspace{0.1cm} \int\limits^{5}_{1}c(x-1)dx \hspace{0.1cm}+\hspace{0.1cm} \int\limits^{\infin}_{5}0dx1\\ \Rightarrow 0 \hspace{0.1cm}+\hspace{0.1cm} c \big [\frac{x^2}{2}- x\big]^5_1 +0 =1\\ \Rightarrow c\big [\frac{x^2}{2}- x\big]^5_1\\ \Rightarrow 8c = 1\\ \Rightarrow c = \frac{1}{8}

Example 3: If the probability density function is given as: \bold{f(x)= \begin{cases} \frac{5}{2}x^2 & 0\leq x < 2\\ 0 & otherwise \end{cases}} . Find the mean.

Solution:

Formula for mean:

μ = \int\limits^{\infin}_{-\infin}xf(x)dx

⇒ μ = \int\limits^{1}_{-\infin}x(0) dx\hspace{0.1cm}+\hspace{0.1cm} \int\limits^{2}_{1} x(\frac{5x^2}{2}) dx \hspace{0.1cm}+\hspace{0.1cm} \int\limits^{\infin}_{2}x(0) dx

⇒ μ = \frac{5}{2}\big[\frac{x^4}{4}\big]^2_1

⇒ μ = (5/2) × (15/4)

⇒ μ = 75/8 = 9.375

Example 4: If the probability density function is given as: \bold{f(x)= \begin{cases} {2}{}x & 0\leq x < 1\\ 0 & otherwise \end{cases}} . verify if this is a valid probability density function.

To verify that f(x) is a valid PDF, it must satisfy two conditions:

f(x)≥0 for all x.

The integral of f(x) over its entire range must equal 1.

Checking f(x)≥0:

f(x)=2x is clearly non-negative for 0≤x≤10.

Integrating f(x) over its range:∫−∞∞f(x) dx=∫012x dx=[x2]01=12−02=1.\int_{-\infty}^{\infty} f(x) \, dx = \int_{0}^{1} 2x \, dx = \left[ x^2 \right]_{0}^{1}

= 1² - 0² = 1.

Since both conditions are satisfied, f(x) is a valid PDF.

Example 5: Given the probability density function f(x)= \begin{cases} 3x^2 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}, find the mean (expected value) of the distribution.

The mean of a continuous random variable X with PDF f(x) is given by:

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

For the given PDF:

E(X)= = \int_{0}^{1} x \cdot 3x^2 \, dx

= \int_{0}^{1} 3x^3 \, dx

= 3 \left[ \frac{x^4}{4} \right]_{0}^{1}

= 3 \cdot \frac{1}{4}

= \frac{3}{4}

Example 6: Using the same PDF f(x) = \begin{cases} 3x^2 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}​, find the variance of the distribution.

The variance of a continuous random variable X is given by:

\text{Var}(X) = E(X^2) - [E(X)]^2.

We already have E(X) = \frac{3}{4}​.

Now, we need to find E(X²):

E(X^2) = \int_{-\infty}^{\infty} x^2 f(x) \, dx

= \int_{0}^{1} x^2 \cdot 3x^2 \, dx

= \int_{0}^{1} 3x^4 \, dx

= 3 \left[ \frac{x^5}{5} \right]_{0}^{1}

= 3 \cdot \frac{1}{5}

= \frac{3}{5}.

Practice Questions on Probability Density Function

Q 1: Let f(x) be a probability density function given by:

• f(x) = 2x for 0 ≤ x ≤ 2
• f(x) = 0 otherwise

Verify that f(x) is a valid probability density function.

Q 2: Let f(x) be a probability density function given by:

• f(x) = 1/2e^-x/2 for x ≥ 0
• f(x) = 0 for x < 0

Calculate the probability that X ≤ 1.

Q 3: Let f(x) be a probability density function given by:

• f(x) = 2x² for 0 ≤ x ≤ 1
• f(x) = 0 otherwise

Find the cumulative distribution function (CDF) F(x) for x ≥ 0.

Q 4: Given the probability density function f(x) of a continuous random variable X:

• f(x) = 3(1 - x²) for -1 ≤ x ≤ 1
• f(x) = 0 otherwise

Find P(0 ≤ X ≤ 1/2)

Q 5: Find the cumulative distribution function (CDF) for the PDF f(x) = \begin{cases} 2x & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}​.

Q 6: Given the function f(x) = \begin{cases} k(1-x^2) & \text{if } -1 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}​, find the value of k that makes f(x) a valid PDF.

Q 7: Using the same PDF f(x) = \begin{cases} \frac{1}{3} e^{-x/3} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases}otherwise​, find the variance Var(X).

Q 8: Given the PDF f(x) = \begin{cases} 3(1-x)^2 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}​, find the cumulative distribution function (CDF) F(x).

Q 9: For the PDF f(x) = \begin{cases} \frac{5}{4}(x - x^2) & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}, calculate the probability that X is between 0.2 and 0.8, i.e., P(0.2 \leq X \leq 0.8).

Q 10: For the PDF f(x) = \begin{cases} \frac{1}{3} e^{-x/3} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases}​, calculate the expected value E(X).