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## 4.1.4 Solved Problems: Continuous Random Variables

- Find the constant $c$.
- Find $EX$ and Var$(X)$.
- Find $P(X \geq \frac{1}{2})$.
- To find $P(X \geq \frac{1}{2})$, we can write $$P(X \geq \frac{1}{2})=\frac{3}{2} \int_{\frac{1}{2}}^{1} x^2dx=\frac{7}{16}.$$

Let $X$ be a positive continuous random variable. Prove that $EX=\int_{0}^{\infty} P(X \geq x) dx$.

## Probability Density Function

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14.1 - probability density functions.

## Example 14-1 Section

Such a curve is denoted \(f(x)\) and is called a (continuous) probability density function .

\(f(x)\) is positive everywhere in the support \(S\), that is, \(f(x)>0\), for all \(x\) in \(S\)

The area under the curve \(f(x)\) in the support \(S\) is 1, that is:

## Example 14-2 Section

Let \(X\) be a continuous random variable whose probability density function is:

Now, let's first start by verifying that \(f(x)\) is a valid probability density function.

What is \(P\left(X=\frac{1}{2}\right)\)?

It is a straightforward integration to see that the probability is 0:

\(\int^{1/2}_{1/2} 3x^2dx=\left[x^3\right]^{x=1/2}_{x=1/2}=\dfrac{1}{8}-\dfrac{1}{8}=0\)

\(P(a\le X\le b)=P(a<X\le b)=P(a\le X<b)=P(a<x<b)\)

for any constants \(a\) and \(b\).

## Example 14-3 Section

## Probability Density Function

## What is Probability Density Function?

## Probability Density Function Definition

## Probability Density Function Example

## Probability Density Function Formula

## Probability Density Function of Continuous Random Variable

f(x) = \(\frac{\mathrm{d} F(x)}{\mathrm{d} x}\) = F'(x)

P(a < X ≤ b) = F(b) - F(a) = \(\int_{a}^{b}f(x)dx\)

Here, F(b) and F(a) represent the cumulative distribution function at b and a respectively.

## Probability Density Function of Normal Distribution

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

## Probability Density Function Graph

## Mean of Probability Density Function

E[X] = \(\mu = \int_{-\infty }^{\infty}xf(x)dx\)

## Median of Probability Density Function

\(\int_{-\infty }^{m}f(x)dx = \int_{m}^{\infty }f(x)dx\) = 1/2

## Variance of Probability Density Function

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

Var(X) = \(\sigma ^{2} = \int_{-\infty }^{\infty }(x - \mu )^{2}f(x)dx\)

## Properties of Probability Density Function

- f(x) ≥ 0. This implies that the probability density function for all real numbers can be either equal to or greater than 0. But it can never be negative or lesser than 0.
- \(\int_{-\infty }^{\infty}f(x)dx\) = 1. Thus, the total area under the probability density curve will be equal to 1.

Important Notes on Probability Density Function

- Probability density function determines the probability that a continuous random variable will fall between a range of specified values.
- On differentiating the cumulative distribution function, we obtain the probability density function.
- The mean of the probability density function can be give as E[X] = \(\mu = \int_{-\infty }^{\infty}xf(x)dx\).
- As the median divides the probability density function curve into 2 equal halves, its value will be equal to 1 / 2.
- The variance of a probability density function is given by Var(X) = \(\sigma ^{2} = \int_{-\infty }^{\infty }(x - \mu )^{2}f(x)dx\)

## Examples on Probability Density Function

- Example 1: If the probability density function is given as: f(x) = \(\left\{\begin{matrix} x(x-1) &0\leq x < 3 \\ x& x\geq3 \end{matrix}\right.\) Find P(1< X < 2). Solution: Integrating the function, \(\int_{1}^{2}x(x-1)dx\) = \(\int_{1}^{2}(x^{2}-x)dx\) = \([\frac{x^{3}}{3} - \frac{x^{2}}{2}]_{1}^{2}\) = 5 / 6 Answer: P(1< X < 2) = 5 / 6
- Example 2: If X is a continuous random variable with the probability density function given as: f(x) = \(\left\{\begin{matrix} \frac{be^{-x}}{2} & x\geq 0\\ 0 & \text{otherwise} \end{matrix}\right.\) Find the value of b. Solution: We know from properties of probability density function that \(\int_{-\infty }^{\infty}f(x)dx\) = 1 \(\int_{0}^{\infty }\frac{be^{-x}}{2}dx\) = 1 On integrating, \(\frac{b}{2}\left [ -e^{-x} \right ]_{0}^{\infty }\) = 1 b / 2 = 1 b = 2 Answer: b = 2
- Example 3: Find the expected value of X if the probability density function is defined as: f(x) = \(\left\{\begin{matrix} \frac{3}{2}x^{2} & 0\leq x\leq 2\\ 0& \text{otherwise} \end{matrix}\right.\) Solution: We know that, E[X] = \(\int_{-\infty }^{\infty}xf(x)dx\) = \(\int_{-\infty }^{0}x(0)dx + \int_{0}^{2}x (\frac{3x^{2}}{2})dx + \int_{2}^{\infty }x(0)dx\) = \(\frac{3}{2}[\frac{x^{4}}{4}]_{0}^{2}\) = (3/8) [16] = 6 Answer: Mean = 6

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## Practice Questions on Probability Density Function

## FAQs on Probability Density Function

What is meant by the probability density function.

## What is the Probability Density Function Formula?

## How to Calculate the Probability Density Function?

## How to Find the Mean of Probability Density Function?

## Is Probability Density Function Always Positive?

## How Do You Find the Probability of a Probability Density Function?

## What are the Features of Probability Density Function?

The features of the probability density function are given below:

- The probability density function will always be a positive value.
- The total area under the probability density function curve will always be equal to 1.

## Probability Density Function

## What is Probability Density Function?

The probability density function gives the output indicating the density of a continuous random variable lying between a specific range of values. If a given scenario is calculated based on numbers and values, the function computes the density corresponding to the specified range.

## Table of contents

## Examples with Calculation

Applications, frequently asked questions (faqs), recommended articles, key takeaways.

- The probability density function (PDF) gives the output indicating the density of a continuous random variable lying between a specific range of values.
- There are imperatively two types of variables: discrete and continuous. The PDF turns into the probability mass function when dealing with discrete variables.
- It is used in statistical calculations and graphically represented as a bell curve forming a relationship between the variable and its probability.
- Its application is significant in machine learning algorithms, analytics, probability theory, neural networks, etc.

## Probability Density Function Explained

Probability density function formula :

f(x) is the PDF and F(x) is the CDF

- X lies between lower limit ‘a’ and upper limit ‘b’
- F(b): Cumulative distribution function at a
- F(a): Cumulative distribution function at b

To better understand the formula and its application, consider the following PDF example:

Applying the upper and lower limits:

- It is used in machine learning algorithms, analytics, probability theory, neural networks, etc.
- Calculates probabilities associated with continuous random variables.
- The PDF of stock price returns is used for studying various market scenarios.
- It is used to model various processes and derive solutions to the problem. For example, it is applied to model chemically reacting turbulent flows and derive appropriate resolution to the closure problems.

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## Probability mass function and probability density function

If \(X\) is a discrete random variable, then use a probability mass function, which is a summation.

## Probability density function graph

First of all, what is a probability density function?

\(\displaystyle \int_X f_X(x) \, \mathrm{d} x = 1\).

That looks more complicated than it actually is. Let's relate it to the graph of a function.

\[ f_X(x) = \begin{cases} 0.1 & 1 \le x \le 11 \\ 0 & \text{otherwise} \end{cases} \]

## Probability density function properties

Using the definition of the probability density function, you can see an important property of them:

It also doesn't matter if you use strict inequalities with continuous density functions:

Both of those properties come from the fact that

\[ P(a<X<b) = \int_a^b f_X(x) \, \mathrm{d} x .\]

\[ f_X(x) = \begin{cases} 2 & 0 \le x \le \dfrac{1}{2} \\ 0 & \text{otherwise} \end{cases} .\]

## Probability density function of normal distribution

## Probability density function example

Suppose that someone tells you that

\[ f_X(x) = \begin{cases} 2x & 0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases}\]

(a) Check to be sure this is a probability density function.

(b) Find the probability you will wait less than half an hour to see the doctor.

(c) Find the probability you will wait more than half an hour to see the doctor.

So this is, in fact, a probability density function.

\[ P(X > 0.5) = 1 - P(X<0.5).\]

## Probability Density Function - Key takeaways

## Frequently Asked Questions about Probability Density Function

The PDF is the probability density function and the CDF is the cumulative distribution function.

## --> What is probability density function give example?

## --> Can probability density function be negative?

No. They are always at least zero.

## --> What does the probability density function tell us?

## --> Can a probability density function be greater than 1?

## Final Probability Density Function Quiz

Probability density functions are used with ____ random variables.

\(f_X(x)\) is always at least zero.

The area under the curve is equal to \(1\).

True or False: For a continuous random variable \(X\), \( P(a\le X \le b) = P(a<X <b)\).

\( P(a<X<b) = \displaystyle\int_a^b f_X(x) \, \mathrm{d} x\).

\[ f(x) = \begin{cases} 1 & 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}\]

a probability density function for a continuous random variable \(X\)?

No. The area under the curve of \(f(x)\) is not equal to \(1\).

\[ f(x) = \begin{cases} 2 & 0 \le x \le \frac{1}{2} \\ 0 & \text{otherwise} \end{cases}\]

Yes. It satisfies all three properties of a probability density function.

\[ f(x) = \begin{cases} \frac{1}{2} & 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}\]

For a probability density function, what is \(P(X=a)\)?

For a probability mass function, what is \(P(X=a)\)?

Not enough information to tell.

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## Section 8.5 : Probability

- Show that \(f\left( x \right)\) is a probability density function.
- Find \(P\left( {X \le 7} \right)\).
- Find \(P\left( {X \ge 7} \right)\).
- Find \(P\left( {3 \le X \le 14} \right)\).
- Determine the mean value of \(X\).
- Verify that \(f\left( t \right)\) is a probability density function.
- What is the probability that a light bulb will have a life span less than 8 months?
- What is the probability that a light bulb will have a life span more than 20 months?
- What is the probability that a light bulb will have a life span between 14 and 30 months?
- Determine the mean value of the life span of the light bulbs.
- Determine the value of \(c\) for which the function below will be a probability density function. \[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\left( {8{x^3} - {x^4}} \right)}&{{\mbox{if }}0 \le x \le 8}\\0&{{\mbox{otherwise}}}\end{array}} \right.\] Solution

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Problem. Let X be a continuous random variable with PDF given by fX(x)=12e−|x|,for all x∈R. If Y=X2, find the CDF of Y. Solution.

a) Verify that f is a probability density function. b) What is the probability that x is greater than 4. c) What is the probability that x is between 1 and 3

Solution. In reality, I'm not particularly interested in using this example just so that you'll know whether or not you've been ripped

In this video lecture you will know the relationship between probability and probability density function (PDF). This problem on probability

Say we have a continuous random variable whose probability density function is given by f(x) = x + 2, when 0 < x ≤ 2. We want to find P(0.5 < X < 1). Then we

Let the joint probability density function for (X, Y) be f(x, y) = 2 yx. +. , x > 0, y > 0,. 3x + y < 3, zero otherwise. a). Find the probability P(X < Y).

Consider an example with PDF, f(x) = x + 3, when 1 < x ≤ 3. We have to find P(2 < X < 3). Integrating x + 3 within the limits 2 and 3 gives the answer 5.5.

An example of a probability density function for a continuous random variable would be the standard normal distribution. Can probability density function be

Determine the mean value of X X . Solution; For a brand of light bulb the probability density function of the life span of the light bulb is

The lifetime of a certain brand of battery, in tens of hours, is modelled by the continuous random variable X with probability density function ( ).