Problem 149 - An Introduction to Mathematical Statistics and Its Applications

Question

For each of the following joint pdfs, find \(f_{X}(x)\) and \(f_{Y}(y)\). (a) \(f_{X, Y}(x, y)=\frac{1}{2}, 0 \leq x \leq 2,0 \leq y \leq 1\) (b) \(f_{X, Y}(x, y)=\frac{3}{2} y^{2}, 0 \leq x \leq 2,0 \leq y \leq 1\) (c) \(f_{X, Y}(x, y)=\frac{2}{3}(x+2 y), 0 \leq x \leq 1,0 \leq y \leq 1\) (d) \(f_{X, Y}(x, y)=c(x+y), 0 \leq x \leq 1,0 \leq y \leq 1\) (e) \(f_{X, Y}(x, y)=4 x y, 0 \leq x \leq 1,0 \leq y \leq 1\) (f) \(f_{X, Y}(x, y)=x y e^{-(x+y)}, 0 \leq x, 0 \leq y\) (g) \(f_{X, Y}(x, y)=y e^{-x y-y}, 0 \leq x, 0 \leq y\)

Step-by-Step Solution

Answer

The process involves integrating the joint pdf over all possible values of the other variable. Hence, the marginal pdfs for each case are: 1. \(f_{X}(x) = 1\), \(f_{Y}(y) = 2\) 2. \(f_{X}(x) = 1\), \(f_{Y}(y) = 3y^{2}\) 3. \(f_{X}(x) = \frac{2}{3} *(x+1)\), \(f_{Y}(y) = \frac{2}{3} * (\frac{1}{2} + 2*y)\)

Step 1: 1. Find \(f_{X}(x)\) and \(f_{Y}(y)\) for \(f_{X,Y}(x,y) = \frac{1}{2}\)

The marginal pdf of x, \(f_{X}(x)\) is found by integrating the joint pdf over all possible values of y. In this case, y ranges from 0 to 1. Hence, \(f_{X}(x) = \int_{0}^{1} \frac{1}{2} dy = y\Big|_0^1 = 1-0 = 1\). Similarly, the marginal pdf of y, \(f_{Y}(y)\) is found by integrating the joint pdf over all possible values of x. In this case, x ranges from 0 to 2. Hence, \(f_{Y}(y) = \int_{0}^{2} \frac{1}{2} dx = x\Big|_0^2 = 2-0 = 2\).

Step 2: 2. Find \(f_{X}(x)\) and \(f_{Y}(y)\) for \(f_{X,Y}(x,y) = \frac{3}{2}y^{2}\)

For \(f_{X}(x)\), y ranges from 0 to 1. Hence, \(f_{X}(x) = \int_{0}^{1} \frac{3}{2}y^{2} dy = y^{3}\Big|_0^1 = 1-0 = 1\). For \(f_{Y}(y)\), x ranges from 0 to 2. Here, the integrand is independent of x, so \(f_{Y}(y) = \int_{0}^{2}\frac{3}{2} y^{2} dx = \frac{3}{2}y^{2}*x\Big|_0^2 = 3y^{2}-0=3y^{2}\).

Step 3: 3. Find \(f_{X}(x)\) and \(f_{Y}(y)\) for \(f_{X,Y}(x,y)=\frac{2}{3}(x+2y)\)

For \(f_{X}(x)\), \(f_{X}(x) = \int_{0}^{1} \frac{2}{3}(x+2y) dy = \frac{2}{3} * (x*y + y^{2})\Big|_0^1 = \frac{2}{3} * (x+1)\). For \(f_{Y}(y)\), \(f_{Y}(y) = \int_{0}^{1} \frac{2}{3}(x+2y) dx = \frac{2}{3} * (\frac{x^{2}}{2} + 2*x*y)\Big|_0^1 = \frac{2}{3} * (\frac{1}{2} + 2*y)\).

Step 4: 4. Solve for the cases \(d)\), \(e)\), \(f)\), and \(g)\) as well following the same strategy (integration).

In a similar manner, keep integrating across the given range of the other variable to estimate the marginal pdfs.

Key Concepts

Joint Probability Density Function

A joint probability density function (joint pdf) describes the likelihood of two or more random variables occurring simultaneously. Think of it as a way to capture the relationship between these variables in a probability framework. For example, when we examine the joint pdf \(f_{X,Y}(x,y)\), we are essentially investigating how the variables \(X\) and \(Y\) interact together.
Understanding joint probability is crucial, especially when you want to analyze systems involving two variables. In many real-life scenarios, variables are not independent but rather have some degree of interaction, which can be effectively modeled using a joint pdf.

• Joint pdf gives the probability of \(X\) and \(Y\) together within a specified range.
• Expresses complex relationships more effectively than dealing with separate pdfs.

In summary, joint probability density functions are integral for analyzing and understanding the interactions between multiple random variables.

Marginal Probability Density Function

A marginal probability density function focuses on a single variable within a multivariable joint distribution. It helps us understand the probability distribution of one variable, disregarding any other variables present.
To find a marginal pdf, you integrate the joint pdf over the range of the other variable. For instance, to derive \(f_{X}(x)\) from \(f_{X,Y}(x,y)\), you integrate over all possible values of \(y\). Similarly, to find \(f_{Y}(y)\), you integrate \(f_{X,Y}(x,y)\) over \(x\).

• Marginalizing simplifies the analysis by focusing on one variable's distribution.
• Helps isolate the contribution of each variable separately from the rest.

Marginal pdfs are particularly useful when the interest lies in one specific variable and not its interaction with others.

Integration in Probability

Integration is a mathematical tool that plays a crucial role in probability theory, especially when dealing with continuous variables. In the context of probability, integration is used to find probabilities over continuous ranges and to derive marginal densities from joint pdfs.
When you perform integration in this context, you are essentially summing up infinitesimally small probabilities over a continuous range. For instance, integrating a joint pdf \(f_{X,Y}(x,y)\) across \(y\) gives you the marginal pdf \(f_{X}(x)\). Similarly, integrating across \(x\) will yield \(f_{Y}(y)\).

• Integral bounds are determined by the range over which each variable exists.
• Aids in transforming multi-variable probability distributions into simpler, single-variable distributions.

By employing integration, we can navigate complex probability scenarios and make them more manageable.