Problem 160 - An Introduction to Mathematical Statistics and Its Applications

Question

Find the joint pdf associated with two random variables \(X\) and \(Y\) whose joint cdf is $$ F_{X, Y}(x, y)=\left(1-e^{-\lambda y}\right)\left(1-e^{-\lambda x}\right), \quad x>0, \quad y>0 $$

Step-by-Step Solution

Answer

The joint probability density function (pdf) associated with the random variables \(X\) and \(Y\) is \(f_{X,Y}(x,y) = \lambda^2 e^{-\lambda y} e^{-\lambda x}\), for \(x>0, y>0\).

Step 1: Understand the Problem

The problem provides the joint Cumulative Distribution Function (CDF) of two random variables X and Y, and it is required to find the corresponding joint Probability Density Function (PDF). Recall that the CDF describes the probability that random variables X and Y take on a value less than or equal to x and y respectively, while the PDF describes the likelihood of random variables X and Y taking on a specific value.

Step 2: Obtain the Derivative

To get the joint PDF, it is necessary to take the derivative of joint CDF with respect to X and Y successively. Let's begin with the joint CDF \(F_{X, Y}(x, y)=(1-e^{-\lambda y})(1-e^{-\lambda x})\), for \(x>0, y>0\), thus the joint PDF is obtained by differentiating the joint CDF with respect to X and Y, i.e., \(f_{X,Y}(x,y)= \frac{\partial^2}{\partial x\partial y} F_{X, Y}(x, y)\).

Step 3: Calculate the Partial Derivatives

First, let's take the partial derivative with respect to x: \(f_{X}(x) = \frac{\partial}{\partial x} F_{X, Y}(x, y) = \lambda e^{-\lambda x}(1-e^{-\lambda y})\). Then, take the partial derivative of \(f_{X}(x)\) with respect to y: \(f_{X,Y}(x,y) = \frac{\partial}{\partial y} f_{X}(x) = \lambda^2 e^{-\lambda y} e^{-\lambda x}\).

Key Concepts

Joint Cumulative Distribution Function

The joint cumulative distribution function (joint CDF) of two random variables provides the probability that both variables take on a value less than or equal to a given pair of numbers. For our example, the joint CDF is given by the formula:\[F_{X, Y}(x, y)=\left(1-e^{-\lambda y}\right)\left(1-e^{-\lambda x}\right)\]This formula represents the cumulative probability for the variables \(X\) and \(Y\) when both are greater than zero. In simpler terms, it calculates how likely it is for \(X\) to be less than or equal to certain value \(x\) and \(Y\) to be less than or equal to a certain value \(y\) simultaneously.
In practical terms, understanding the joint CDF becomes crucial in multi-variable analyses where the combined behavior of two variables is of interest. It combines the probabilities of each random variable into a single function, making it helpful in predicting and modeling real-world situations where two or more outcomes are interdependent.

Partial Derivatives

Partial derivatives are a crucial mathematical tool used to understand how a function changes as one of its variables changes, while holding the other variables constant. To find the joint probability density function (PDF) from a joint cumulative distribution function (CDF), we utilize partial derivatives.
The joint PDF is obtained by computing the second partial derivative of the joint CDF with respect to each variable, \(X\) and \(Y\). In our example:

• First, the partial derivative of the joint CDF with respect to \(X\) is computed:
  \[ \frac{\partial}{\partial x} F_{X, Y}(x, y) = \lambda e^{-\lambda x}(1-e^{-\lambda y}) \]
• Next, the partial derivative of this result with respect to \(Y\) is taken:
  \[ \frac{\partial}{\partial y} f_{X}(x) = \lambda^2 e^{-\lambda y} e^{-\lambda x} \]

This process transforms the joint CDF into the joint PDF. The partial derivatives isolate the rate of change for each variable independently, offering insights into how each dimension affects the joint probability.

Random Variables

Random variables are fundamental components in the study of probability and statistics. A random variable is a numerical value that is determined by the outcome of a random phenomenon. They are essential in modeling and analyzing real-world situations where outcomes are uncertain.
In considering two random variables, \(X\) and \(Y\), we can study them individually as separate entities or jointly, as in this problem. Joint analysis of random variables allows us to comprehend their collective statistical behavior.
Key points about random variables:

• The random variable \(X\) might represent a specific measurement like time or distance.
• Likewise, \(Y\) can represent another related variable, and together, their joint distribution can model situations where the variables influence each other.
• Understanding the properties of random variables, such as their mean and variance, is crucial for effective probabilistic modeling.

Overall, by examining their joint distributions, we gain a comprehensive understanding of how random variables interact in varied scenarios, enhancing our ability to predict and analyze complex systems.