Problem 10 - Applied Probability

Question

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=1\) for \(0 \leq t \leq 1,0 \leq u \leq 2(1-t)\) \(P(X>1 / 2, Y>1), \quad P(0 \leq X \leq 1 / 2, Y>1 / 2), P(Y \leq X)\)

Step-by-Step Solution

Answer

The marginal density functions are \(f_{X}(x) = 2(1-x)\) for \(0 \leq x \leq 1\) and \(f_{Y}(y) = 1 - \frac{y}{2}\) for \(0 \leq y \leq 2\). Analytical probabilities can be found by integrating the joint PDF over specified limits. Discrete approximations require summing the areas of rectangles that fall within the desired regions on the joint PDF's support.

Step 1: Sketch the Region of Definition

The given joint PDF is defined for the region where \(0 \leq t \leq 1\) and \(0 \leq u \leq 2(1-t)\). Sketch this region on the XY-plane. This will form a triangle with vertices at points (0,0), (1,0), and (0,2).

Step 2: Determine Marginal Density \(f_X\)

To find the marginal density \(f_{X}(x)\), integrate the joint PDF \(f_{XY}(x, y)\) over the range of \(y\): \(f_{X}(x) = \int_{0}^{2(1-x)} 1 \, dy\).

Step 3: Solve for \(f_X\)

Evaluating the integral from Step 2 gives: \(f_{X}(x)=y|_{0}^{2(1-x)}\), resulting in \(f_{X}(x)=2(1-x)\) for \(0 \leq x \leq 1\).

Step 4: Determine Marginal Density \(f_Y\)

To find the marginal density \(f_{Y}(y)\), integrate the joint PDF \(f_{XY}(x, y)\) over the range of \(x\): \(f_{Y}(y) = \int_{0}^{1-\frac{y}{2}} 1 \, dx\).

Step 5: Solve for \(f_Y\)

Evaluating the integral from Step 4 gives: \(f_{Y}(y)=x|_{0}^{1-\frac{y}{2}}\), resulting in \(f_{Y}(y)=1-\frac{y}{2}\) for \(0 \leq y \leq 2\).

Step 6: Discrete Approximation and Plot of \(f_X\)

To approximate \(f_{X}(x)\), divide the interval \([0,1]\) into n subintervals. Calculate the function values at these points, and plot these to estimate the shape of the marginal density.

Step 7: Plot the Marginal Distribution Function \(F_X\)

Use the values calculated for \(f_{X}(x)\) in the discrete approximation and cumulatively sum these to estimate \(F_{X}(x)\), the marginal distribution function, then plot it.

Step 8: Calculate Analytical Probabilities

Evaluate \(\int_{1/2}^{1} \int_{1}^{2(1-t)} f_{XY}(t,u)\, du\, dt\), \(\int_{0}^{1/2} \int_{1/2}^{2(1-t)} f_{XY}(t,u)\, du\, dt\), and \(\int_{0}^{1} \int_{0}^{t} f_{XY}(t,u)\, du\, dt\) to find the probabilities for \(P(X>1/2, Y>1)\), \(P(0 \leq X \leq 1/2, Y>1/2)\), and \(P(Y \leq X)\), respectively.

Step 9: Discrete Approximation of Probabilities

Divide the region of definition into a grid with small rectangles and sum up the areas of those rectangles that fall within the regions corresponding to the probabilities required. This will approximate \(P(X>1/2, Y>1)\), \(P(0 \leq X \leq 1/2, Y>1/2)\), and \(P(Y \leq X)\). Each rectangle has an area equal to the rectangle's width times height times the value of the PDF at that location.

Key Concepts

Joint Probability Distribution Function

In probability theory, the joint probability distribution function (PDF) showcases the probability that two random variables, typically denoted as X and Y, will take on a specific pair of values. This function provides a complete description of the probability structure for the pair of variables, letting us understand the likelihood of different outcomes occurring together.

The joint PDF is useful in applications where variables depend on each other. For example, in our exercise, the joint PDF is specified as a simple constant value within a certain region, indicating that all points within this region are equally likely. Analyzing such a function involves sketching the region where it's defined, which often provides insights regarding the dependencies and potential correlations between the variables.

Integration of Probability Density Functions

When dealing with continuous random variables, the integration of probability density functions becomes a fundamental operation in computing numerous important quantities, such as marginal densities and probabilities of finding a random variable within a certain interval.

Integration is essentially a way to sum probabilities over a continuous interval, and in the case of joint PDFs, we integrate over one variable to find the marginal PDF of the other. This process is highlighted in our exercise where integrating the joint PDF of X and Y over the range of one variable gives us the marginal density function of the other. This is key in transforming a joint PDF into its marginal components, isolating the individual behavior of each random variable.

Marginal Distribution Function

Moving beyond how variables interact jointly, the marginal distribution function captures the probability distribution of a single random variable regardless of the value of another associated variable. It is derived from the joint distribution by integrating out the other variable, as seen in the step-by-step solution where the marginal densities for X and Y are calculated.

Understanding marginal distributions is essential in probability theory for examining the behavior of individual variables when multiple random factors are at play. The exercise also exemplifies how to plot the marginal distribution using discrete approximation, which is important for visualizing probabilities and gaining deeper insights into the distribution of a single variable within a dataset.

Probability Theory

At the crux of our exercise, and many others like it, lies probability theory. This mathematical framework is designed to study randomness and uncertainty, providing the tools to analyze events and their likelihood. It incorporates the principles and methods used to assess the probability of events resulting from random processes.

Probability theory lays the groundwork for defining concepts such as random variables, their expected values, variances, and crucially, their distributions—whether joint, marginal, or conditional. It allows for the mathematical expression of uncertainty and the quantification of the stochastic relationships between events, as illustrated by the various probabilities calculated analytically and through discrete approximation in our step-by-step solutions.

Discrete Approximation

Discrete approximation is a numerical method used to estimate the values of functions or probabilities when analytical solutions are complex or impossible to find. In probability theory, it can be particularly useful for approximating the areas under a curve to estimate probabilities, as demonstrated in the solutions for our exercise.

By dividing the area under the function into small, discrete parts—such as rectangles or trapezoids—we can approximate integration. While it might not give an exact result, with enough subdivisions, the approximation can become very close to the true value. Most importantly, it provides a practical approach for visualizing and understanding probability distributions, particularly so for students who are just getting familiar with the concepts.