Problem 173 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that random variables \(X\) and \(Y\) are independent with marginal pdfs \(f_{X}(x)=2 x, 0 \leq x \leq 1\), and \(f_{Y}(y)=3 y^{2}, 0 \leq y \leq 1\). Find \(P(Y

Step-by-Step Solution

Answer

The probability \(P(Y

Step 1: Interpret the independent random variables

Given that \(X\) and \(Y\) are independent random variables, this means that the event of one variable occurring will not affect the other. In this context, the joint pdf \(f(x, y)\) will be equal to the product of the marginal pdfs.

Step 2: Calculate the joint pdf

The joint pdf of the two independent random variables \(X\) and \(Y\) is given by the product of their marginal pdfs. Therefore, we have \(f_{X, Y}(x, y) = f_{X}(x) * f_{Y}(y) = 2x * 3y^2 = 6xy^2\).

Step 3: Evaluate the probability

The probability that \(Y

Step 4: Evaluate the integral

By integrating step by step, first with respect to \(y\) and then with respect to \(x\), we obtain: \[P(Y

Key Concepts

Joint Probability Density Function

Understanding the joint probability density function (pdf) is essential when dealing with two or more continuous random variables. In essence, the joint pdf, denoted as \( f(x, y) \) for variables \( X \) and \( Y \), represents the probability that \( X \) falls within a particular range and \( Y \) falls within another, simultaneously. When variables \( X \) and \( Y \) are independent, as in the exercise, their joint pdf is the product of their individual marginal pdfs.

This attribute of independent random variables greatly simplifies calculations, since it means the behavior of one variable does not influence or alter the behavior of the other. For \( X \) and \( Y \), this results in a joint pdf of \( f_{X,Y}(x, y) = f_{X}(x) \cdot f_{Y}(y) \).

In practical terms, if you're presented with a scenario where you need to find the likelihood of both variables falling within specific bounds, the joint pdf is the function you'd use to compute that probability. It facilitates the evaluation of the system as a whole rather than considering each variable in isolation. The principle is straightforward for independent variables: find their individual marginal pdfs, multiply them together, and voila, you have the joint pdf.

Double Integral Probability

The concept of double integral probability is a fundamental tool in computing probabilities over continuous variables. It becomes particularly pertinent when dealing with joint probability density functions, as highlighted in the exercise.

The double integral, symbolized by \( \int \), is essentially a way to sum up all the infinitesimal elements of probability across a two-dimensional region. Translating that to our scenario, the double integral tells us how to find the probability that the value of \( Y \) is less than \( X \) over the specified ranges for each variable.

To compute it, we designate the bounds of integration based on the condition we're interested in—here, \( y < x \). We start by integrating the joint pdf with respect to \( y \), from 0 to \( x \), and then integrate the resulting expression with respect to \( x \), from 0 to 1. The solution reflects the accumulated probability over the designated region, which, when we talk about randomness and chance, equates to our sought-after likelihood or probability.

Marginal Probability Density Functions

The notion of marginal probability density functions (pdfs) allows us to focus on a single variable within a multivariable setup. These marginal pdfs quantify the probability distribution of a subset of variables, disregarding the presence of others.

In the context of our exercise, we dealt with two marginal pdfs, \( f_{X}(x) \) and \( f_{Y}(y) \), describing the probabilities of the independent variables \( X \) and \( Y \), respectively. These functions show the probability that \( X \) or \( Y \) will take on a value within a particular range, without consideration for the value of the other variable.

• \( f_{X}(x) = 2x \text{ for } 0 \leq x \leq 1 \)
• \( f_{Y}(y) = 3y^{2} \text{ for } 0 \leq y \leq 1 \)

By integrating a marginal pdf over its entire range, we would calculate the total probability (which must equal 1, given that the variable must take some value within that range). Marginal pdfs are generally derived from the joint pdf, particularly when variables are not independent. However, in the case of independent variables, one can start with the marginals and construct the joint pdf, as we did in this exercise.

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Problem 172

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Problem 174

Other exercises in this chapter

Problem 171

Let \(X\) and \(Y\) be random variables with joint pdf $$ f_{X, Y}(x, y)=k, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1, \quad 0 \leq x+y \leq 1 $$

View solution

Problem 172

Are the random variables \(X\) and \(Y\) independent if \(f_{X, Y}(x, y)=\frac{2}{3}(x+2 y), 0 \leq x \leq 1,0 \leq y \leq 1 ?\)

View solution

Problem 174

. Find the joint cdf of the independent random variables \(X\) and \(Y\), where \(f_{X}(x)=\frac{x}{2}, 0 \leq x \leq 2\), and \(f_{Y}(y)=2 y, 0 \leq y \leq 1 .

View solution

Problem 175

If two random variables \(X\) and \(Y\) are independent with marginal pdfs \(f_{X}(x)=2 x, 0 \leq x \leq 1\), and \(f_{Y}(y)=1\), \(0 \leq y \leq 1\), calculate

View solution

Key Concepts

Joint Probability Density Function

Understanding the joint probability density function (pdf) is essential when dealing with two or more continuous random variables. In essence, the joint pdf, denoted as \( f(x, y) \) for variables \( X \) and \( Y \), represents the probability that \( X \) falls within a particular range and \( Y \) falls within another, simultaneously. When variables \( X \) and \( Y \) are independent, as in the exercise, their joint pdf is the product of their individual marginal pdfs.

This attribute of independent random variables greatly simplifies calculations, since it means the behavior of one variable does not influence or alter the behavior of the other. For \( X \) and \( Y \), this results in a joint pdf of \( f_{X,Y}(x, y) = f_{X}(x) \cdot f_{Y}(y) \).

In practical terms, if you're presented with a scenario where you need to find the likelihood of both variables falling within specific bounds, the joint pdf is the function you'd use to compute that probability. It facilitates the evaluation of the system as a whole rather than considering each variable in isolation. The principle is straightforward for independent variables: find their individual marginal pdfs, multiply them together, and voila, you have the joint pdf.

Double Integral Probability

The concept of double integral probability is a fundamental tool in computing probabilities over continuous variables. It becomes particularly pertinent when dealing with joint probability density functions, as highlighted in the exercise.

The double integral, symbolized by \( \int \), is essentially a way to sum up all the infinitesimal elements of probability across a two-dimensional region. Translating that to our scenario, the double integral tells us how to find the probability that the value of \( Y \) is less than \( X \) over the specified ranges for each variable.

To compute it, we designate the bounds of integration based on the condition we're interested in—here, \( y < x \). We start by integrating the joint pdf with respect to \( y \), from 0 to \( x \), and then integrate the resulting expression with respect to \( x \), from 0 to 1. The solution reflects the accumulated probability over the designated region, which, when we talk about randomness and chance, equates to our sought-after likelihood or probability.

Marginal Probability Density Functions

The notion of marginal probability density functions (pdfs) allows us to focus on a single variable within a multivariable setup. These marginal pdfs quantify the probability distribution of a subset of variables, disregarding the presence of others.

In the context of our exercise, we dealt with two marginal pdfs, \( f_{X}(x) \) and \( f_{Y}(y) \), describing the probabilities of the independent variables \( X \) and \( Y \), respectively. These functions show the probability that \( X \) or \( Y \) will take on a value within a particular range, without consideration for the value of the other variable.

• \( f_{X}(x) = 2x \text{ for } 0 \leq x \leq 1 \)
• \( f_{Y}(y) = 3y^{2} \text{ for } 0 \leq y \leq 1 \)

By integrating a marginal pdf over its entire range, we would calculate the total probability (which must equal 1, given that the variable must take some value within that range). Marginal pdfs are generally derived from the joint pdf, particularly when variables are not independent. However, in the case of independent variables, one can start with the marginals and construct the joint pdf, as we did in this exercise.