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15.6. Worked Example (Continuous Case)

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Session 1: Introduction to Marginal Distributions

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SarahInstructor

Today, we'll explore marginal distributions, which help us analyze individual random variables from joint probability distributions. Can anyone tell me what a joint probability distribution is?

Noah

Isn't it the probability distribution that involves two or more random variables?

SarahInstructor

Exactly! Now, to find the behavior of a single variable, we can marginalize the joint distribution. Does anyone remember what marginalization is?

Isabella

It's when we integrate out the other variables to focus on one variable.

SarahInstructor

Correct! The whole idea is to simplify complex distributions. Let's move into an example that uses these concepts. So, we have a joint pdf given as... what was that joint pdf?

Akash

It’s 6xy where 0 < x < 1 and 0 < y < 1.

SarahInstructor

Great memory! Now we’ll work on deriving the marginal distributions.

Session 2: Finding Marginal Pdf of X

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RobertInstructor

To find the marginal pdf of X, we integrate the joint pdf over Y. Does anyone remember how we did this?

Ananya

We set up the integral from 0 to 1 of the joint pdf.

RobertInstructor

Exactly! So our equation looks like this: $f_X(x) = \int_0^1 6xy \: dy$ . What comes next?

Noah

We solve the integral of y from 0 to 1!

RobertInstructor

Right! That leads us to $f_X(x) = 3x$ , valid for $0 < x < 1$ . Now, can anyone summarize what that means regarding the probability of X?

Isabella

It gives us the individual probability behavior of X, ignoring Y.

Session 3: Finding Marginal Pdf of Y

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SarahInstructor

Now let's find the marginal pdf for Y using the same method. We again integrate the joint pdf but this time over X. Who can write that out for me?

Akash

The integral should be $f_Y(y) = \int_0^1 6xy \: dx$ .

SarahInstructor

Correct! And what do you find when you compute that integral?

Ananya

It also simplifies to $3y$ for $0 < y < 1$ .

SarahInstructor

Excellent! This shows how each variable can be treated individually. Why is it important to recognize these marginals?

Noah

It helps in understanding each variable's distribution, which can be crucial in engineering applications.

Session 4: Applications of Marginal Distributions

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RobertInstructor

In what fields do you think marginal distributions are used? Let's brainstorm some applications.

Isabella

Signal processing could use them, especially with individual signals.

Akash

Reliability engineering might apply them to estimate failure rates.

RobertInstructor

Great insights! They are also crucial in fields like communication systems and machine learning. Understanding the behavior of individual variables gives us significant analysis power. Can anyone summarize what we’ve learned about marginal distributions?

Ananya

We've learned how to calculate them, their significance, and their applications.

Session 5: Recap and Review

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SarahInstructor

To finish up, can anyone define what we mean by marginal distributions once more?

Noah

It’s the probability distribution of one variable after integrating out others.

SarahInstructor

Perfect! And remember, they’re vital for analyzing systems where we want to focus on specific variables without the complexity of joint distributions. Any questions before we end?

Isabella

Could you remind us about the importance of independent variables again?

SarahInstructor

Of course! If we know X and Y are independent, we can reconstruct the joint pdf as the product of the marginals. That’s a key point! Great work today, everyone!

Overview

Short Summary

This section presents a worked example of finding marginal distributions for continuous random variables.

Medium Summary

The focused worked example guides through calculating the marginal distributions from a given joint probability density function (pdf) for two continuous random variables. It illustrates the determination of marginal pdfs through integration, providing important insights into the behavior of individual variables.

Detailed Summary

Worked Example (Continuous Case)

In this section, we delve into a practical example of finding marginal distributions for two continuous random variables, X and Y, using a specified joint probability density function (pdf). The joint pdf given is:

f_{X,Y}(x,y) = \begin{cases} 6xy, & 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}

The objective is to derive the marginal distributions, $f_X(x)$ and $f_Y(y)$ , from the joint pdf through integration:

Calculating the Marginal Pdf of X: To find the marginal pdf of X, we integrate the joint pdf over all possible values of Y:

f_X(x) = \int_{-\infty}^{\\infty} f_{X,Y}(x,y) , dy = \int_0^1 6xy , dy = 6x \int_0^1 y , dy = 6x \left[\frac{y^2}{2}\right]_0^1 = 3x, \text{ for } 0 < x < 1.

2. **Calculating the Marginal Pdf of Y:** Similarly, for the marginal pdf of Y, we perform the integration over all possible x values:

f_Y(y) = \int_{-\infty}^{\\infty} f_{X,Y}(x,y) , dx = \int_0^1 6xy , dx = 6y \int_0^1 x , dx = 6y \left[\frac{x^2}{2}\right]_0^1 = 3y, \text{ for } 0 < y < 1.

Through this example, we reinforce the understanding and application of marginal distributions. The earnings generated from the calculations detail the separate impacts of each random variable, highlighting their respective probability behaviors independent of one another.

Audio Book

Voice:

Problem Statement

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Problem: Given the joint pdf:

f(x, y) = \begin{cases} 6xy, & 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}

Find the marginal distributions $f(X)$ and $f(Y)$ .

Detailed Explanation

In this problem, we are given a joint probability density function (pdf) for two continuous random variables, X and Y, defined as 6xy within the range (0,1) for both x and y. Our goal is to find the marginal distributions of these two variables, f(X) and f(Y). Marginal distribution focuses on the behavior of a single variable without regard to the other.

Examples & Analogies

Think of this joint pdf as a recipe that tells us how two ingredients, X (e.g., flour) and Y (e.g., sugar), interact in making a cake. If we want to know how much flour to use regardless of how much sugar is in the cake, we look for the marginal distribution (flour) extracted from the overall recipe (joint pdf).

Finding the Marginal pdf of X

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Solution:

Marginal pdf of $X$ :

f(X) = \int_{0}^{1} 6xy \, dy = 6x \int_{0}^{1} y \, dy = 6x \left[ \frac{y^2}{2} \right]_{0}^{1} = 6x \cdot \frac{1^2}{2} = 6x \cdot \frac{1}{2} = 3x$$ Valid for $0 < x < 1$.

Detailed Explanation

To find the marginal pdf of X, we integrate the joint pdf with respect to Y. This process effectively 'removes' the Y variable by summing over its range (from 0 to 1). After evaluating the integral, we find that the marginal pdf for X is 3x, which tells us how the variable X alone behaves within the given limits.

Examples & Analogies

Continuing with the cake analogy, if we want to know just how much flour (X) is needed in the recipe, regardless of the amount of sugar (Y), we look for all combinations of flour and sugar that add up to our cake and determine our total amount of flour needed. This marginal pdf tells us how much of our 'ingredient' is needed independently.

Finding the Marginal pdf of Y

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Marginal pdf of $Y$ :

f(Y) = \int_{0}^{1} 6xy \, dx = 6y \int_{0}^{1} x \, dx = 6y \left[ \frac{x^2}{2} \right]_{0}^{1} = 6y \cdot \frac{1^2}{2} = 6y \cdot \frac{1}{2} = 3y$$ Valid for $0 < y < 1$.

Detailed Explanation

Similarly, we find the marginal pdf for Y by integrating the joint pdf with respect to X. This process reduces the two variables down to just Y. Our solution reveals that the marginal pdf of Y is 3y, indicating how Y behaves on its own without considering X.

Examples & Analogies

Using our recipe analogy again, determining how much sugar (Y) to add on its own, while considering the total contributions of flour is akin to integrating out the effects of flour. The result, the marginal pdf, tells us the necessary amount of sugar independently.

15.6. Worked Example (Continuous Case)

Interactive Audio Lesson

Overview

Short Summary

Medium Summary

Detailed Summary

Worked Example (Continuous Case)

Reference YouTube Videos

partial differential equation lec no 17mp4

Audio Book

Detailed Explanation

Examples & Analogies

Detailed Explanation

Examples & Analogies

Detailed Explanation

Examples & Analogies

Key concepts

Examples

Rhyme

Story

Memory trick

Acronym

Flash Cards

Glossary

Joint Probability Density Function (pdf)

Marginal Distribution

Integration

Probability

Continuous Random Variables

Reference Links

15.6. Worked Example (Continuous Case)

Interactive Audio Lesson

Overview

Short Summary

Medium Summary

Detailed Summary

Worked Example (Continuous Case)

Reference YouTube Videos

partial differential equation lec no 17mp4

Audio Book

Detailed Explanation

Examples & Analogies

Detailed Explanation

Examples & Analogies

Detailed Explanation

Examples & Analogies

Key concepts

Examples

Memory aids

Rhyme

Story

Memory trick

Acronym

Flash Cards

Glossary

Joint Probability Density Function (pdf)

Marginal Distribution

Integration

Probability

Continuous Random Variables

Reference Links