Problem 177 - An Introduction to Mathematical Statistics and Its Applications

Question

Given the joint pdf \(f_{X, Y}(x, y)=2 x+y-2 x y\), \(0

Step-by-Step Solution

Answer

By carefully choosing values for a, b, c, and d and comparing the joint pdf with the product of the marginal pdfs, it can be demonstrated that X and Y are not independent without necessarily computing the exact values of f_X(x) and f_Y(y). The specific choice of a, b, c and d does not affect the independence of X and Y.

Step 1: Compute the Marginal pdfs

The marginal pdf of X can be found by integrating the joint pdf with respect to y over the domain of y. We have \(f_X(x) = \int_0^1 (2 x + y - 2 x y) dy\). Similarly, the marginal pdf of Y is \(f_Y(y) = \int_0^1 (2 x + y - 2 x y) dx\). These integrals need to be computed.

Step 2: Calculate the product of the Marginal pdfs

In this step, compute the product of the marginal pdfs. This will give us \(f_{X}(x) \cdot f_{Y}(y)\).

Step 3: Compare the joint and product of marginal pdfs

Now compare the computed product of the marginal pdfs against the given joint pdf. If these two are not equal, then we can conclude that X and Y are not independent. Additionally, choose specific values for a, b, c and d in a way that the inequalities \(a

Key Concepts

Marginal Probability Density Functions

To understand the behavior of a single random variable within a joint probability distribution, we use marginal probability density functions (pdfs). These functions show the probabilities of the individual random variables irrespective of other variables.

Imagine having two linked features, such as the height and weight of a group of people. The joint probability density function would represent the likelihood of both a specific height and weight occurring together. If we're only interested in one attribute—say, height—the marginal pdf gives the probability of any given height across all weights. This simplification is achieved by 'integrating out' the other variable, essentially summing up all the possibilities of the other variable to focus solely on the one of interest.

For example, if we have a joint pdf \(f_{X, Y}(x, y)\), the marginal pdf for \(X\) is obtained by integrating the joint pdf over the range of \(Y\). In mathematical terms: \(f_X(x) = \int f_{X, Y}(x, y) \, dy\). Similar process is performed to find \(f_Y(y)\) by integrating over the range of \(X\).

This concept is crucial in the study of probability as it allows us to scrutinize the behavior of individual variables in multi-variable contexts.

Independence of Random Variables

Two random variables \(X\) and \(Y\) are independent if the occurrence or value of one does not affect the probability distribution of the other. Independence is a pivotal concept in probability and statistics as it underpins many theoretical models and practical applications.

To determine if \(X\) and \(Y\) from a joint probability distribution are independent, we can compare the joint pdf to the product of the marginal pdfs. If \(f_{X, Y}(x, y) = f_{X}(x) \cdot f_{Y}(y)\) for all values \(x\) and \(y\), then the variables are independent. Conversely, if the equation does not hold, the variables are dependent on each other.

In practical scenarios, testing for independence can involve calculating the probabilities for certain intervals or specific values. Taking subsets of the population, such as looking at people's heights and weights within specific ranges, can reveal dependencies between the variables, which otherwise might be hidden in the overall data distribution.

Integration in Probability

Integration is a key mathematical tool in the field of probability, particularly when working with continuous random variables. It is used to calculate probabilities by summing up infinitesimally small pieces of the probability distribution.

In the context of probability theory, integration allows us to find the probability that a random variable falls within a certain range. This is done by integrating the pdf over that range. The result is a probability value between 0 and 1, indicating how likely it is to observe a value within the specified interval.

The area under the curve of the probability density function gives us the likelihood of the variable taking on values within a certain range. For example, to find the probability that a random variable \(X\) is between \(a\) and \(b\), we would integrate the marginal pdf \(f_X(x)\) from \(a\) to \(b\): \(P(a < X < b) = \int_{a}^{b} f_X(x) \, dx\).

The integral's limits reflect the range of interest, and the integrand is the pdf, which characterizes the likelihood of different outcomes. The integration process in probability is indispensable for determining cumulative distribution functions, expected values, variances, and other important statistical measures.

Previous

Problem 176

Next

Problem 178

Other exercises in this chapter

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

Problem 176

Suppose \(f_{X, Y}(x, y)=x y e^{-(x+y)}, x>0, y>0\). Prove for any real numbers \(a, b, c\), and \(d\) that $$ P(a

View solution

Problem 178

Prove that if \(X\) and \(Y\) are two independent random variables, then \(U=g(X)\) and \(V=h(Y)\) are also independent.

View solution

Problem 179

If two random variables \(X\) and \(Y\) are defined over a region in the \(X Y\)-plane that is not a rectangle (possibly infinite) with sides parallel to the co

View solution

Key Concepts

Marginal Probability Density Functions

To understand the behavior of a single random variable within a joint probability distribution, we use marginal probability density functions (pdfs). These functions show the probabilities of the individual random variables irrespective of other variables.

Imagine having two linked features, such as the height and weight of a group of people. The joint probability density function would represent the likelihood of both a specific height and weight occurring together. If we're only interested in one attribute—say, height—the marginal pdf gives the probability of any given height across all weights. This simplification is achieved by 'integrating out' the other variable, essentially summing up all the possibilities of the other variable to focus solely on the one of interest.

For example, if we have a joint pdf \(f_{X, Y}(x, y)\), the marginal pdf for \(X\) is obtained by integrating the joint pdf over the range of \(Y\). In mathematical terms: \(f_X(x) = \int f_{X, Y}(x, y) \, dy\). Similar process is performed to find \(f_Y(y)\) by integrating over the range of \(X\).

This concept is crucial in the study of probability as it allows us to scrutinize the behavior of individual variables in multi-variable contexts.

Independence of Random Variables

Two random variables \(X\) and \(Y\) are independent if the occurrence or value of one does not affect the probability distribution of the other. Independence is a pivotal concept in probability and statistics as it underpins many theoretical models and practical applications.

To determine if \(X\) and \(Y\) from a joint probability distribution are independent, we can compare the joint pdf to the product of the marginal pdfs. If \(f_{X, Y}(x, y) = f_{X}(x) \cdot f_{Y}(y)\) for all values \(x\) and \(y\), then the variables are independent. Conversely, if the equation does not hold, the variables are dependent on each other.

In practical scenarios, testing for independence can involve calculating the probabilities for certain intervals or specific values. Taking subsets of the population, such as looking at people's heights and weights within specific ranges, can reveal dependencies between the variables, which otherwise might be hidden in the overall data distribution.

Integration in Probability

Integration is a key mathematical tool in the field of probability, particularly when working with continuous random variables. It is used to calculate probabilities by summing up infinitesimally small pieces of the probability distribution.

In the context of probability theory, integration allows us to find the probability that a random variable falls within a certain range. This is done by integrating the pdf over that range. The result is a probability value between 0 and 1, indicating how likely it is to observe a value within the specified interval.

The area under the curve of the probability density function gives us the likelihood of the variable taking on values within a certain range. For example, to find the probability that a random variable \(X\) is between \(a\) and \(b\), we would integrate the marginal pdf \(f_X(x)\) from \(a\) to \(b\): \(P(a < X < b) = \int_{a}^{b} f_X(x) \, dx\).

The integral's limits reflect the range of interest, and the integrand is the pdf, which characterizes the likelihood of different outcomes. The integration process in probability is indispensable for determining cumulative distribution functions, expected values, variances, and other important statistical measures.