Problem 11 - 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. \(\begin{aligned} f_{X Y}(t, u)=4 t &(1-u) \text { for } 0 \leq t \leq 1,0 \leq u \leq 1 \\ & P(1 / 21 / 2), \quad P(X \leq 1 / 2, Y>1 / 2), \quad P(Y \leq X) \end{aligned}\)

Step-by-Step Solution

Answer

The region to sketch is the unit square [0,1]x[0,1], and the marginal density functions are obtained by integrating the joint PDF. To plot the density and distribution via discrete approximation, use sampled points to create histograms or line plots, and calculate analytic probabilities with double integrals. Discrete approximations also rely on point sampling within the specified conditions.

Step 1 - Sketch the Region of Definition

To sketch the region of definition, plot the area in the 2D plane where the joint probability density function (PDF) is defined. The given PDF is defined for the square region where both t (representing X) and u (representing Y) range from 0 to 1. This region is the unit square.

Step 2 - Determine the Marginal Density Function for X

To find the marginal density function \(f_{X}(t)\), integrate the joint PDF \(f_{XY}(t, u)\) with respect to u over the range [0, 1]: \[f_{X}(t) = \int_{0}^{1}4t(1-u)du\]. Perform the integration and simplify.

Step 3 - Determine the Marginal Density Function for Y

Similarly, to find the marginal density function \(f_{Y}(u)\), integrate the joint PDF \(f_{XY}(t, u)\) with respect to t over the range [0, 1]: \[f_{Y}(u) = \int_{0}^{1}4t(1-u)dt\]. Carry out the integration and simplify.

Step 4 - Plot the Marginal Density \(f_{X}\) Using Discrete Approximation

To plot \(f_{X}(t)\) using a discrete approximation, sample points for \(t\) within the range [0, 1] and plot the corresponding values of the marginal density. Use these points to create a histogram or a line plot that approximates \(f_{X}\).

Step 5 - Plot the Marginal Distribution Function \(F_{X}\)

The marginal distribution function \(F_{X}(t)\) can be obtained by integrating the marginal density \(f_{X}(t)\) from 0 to t: \[F_{X}(t) = \int_{0}^{t}f_{X}(u)du\]. Use discrete values of \(t\) to calculate and plot \(F_{X}(t)\) to approximate the distribution function.

Step 6 - Calculate the Indicated Probabilities Analytically

For each of the given probabilities, set up the appropriate double integrals using the joint PDF and evaluate them. For example, for \(P(\frac{1}{2}\frac{1}{2})\), the integral is \[\int_{1/2}^{3/4}\int_{1/2}^{1}4t(1-u)dudt\].

Step 7 - Determine Probabilities by Discrete Approximation

Approximate these probabilities by sampling points from the given region and count how many fall within the specified conditions. For a more accurate approximation, increase the number of sample points.

Key Concepts

Joint Probability Density Function

Understanding the joint probability density function (PDF) is key in the study of how two random variables, often denoted as X and Y, interact.
To visualize a joint PDF like the given function, imagine a surface over a region in a two-dimensional space where each point on the surface represents the likelihood that the random variables take on a specific combination of values. This function is crucial because it lays the groundwork for defining the probabilities of events concerning both X and Y.
In our example, the function is defined within the borders of the unit square, which is the set of all points where both variables lie between 0 and 1, inclusive. This bounded nature of the joint PDF makes it easier to work with, as we only consider a finite region of probability.

Marginal Distribution Function

A marginal density function is what you obtain when you examine just one variable out of the joint PDF.
The process to get there involves integration. For instance, you can find the marginal density function for X, denoted as \( f_{X}(t) \), by integrating the joint PDF over all possible values of Y. This effectively 'margins out' the influence of Y, leaving a function that describes the probability distribution of X alone.
By integrating, we're summing up all the infinitesimally small probabilities across the domain of the other variable, distilling the joint PDF into a function of a single variable. It's like reducing a multilayered cake to just one layer that still retains the flavor of the entire cake.

Discrete Approximation

A discrete approximation steps in when we need to make the continuous world digestible using a finite number of samples. For both plotting and probability calculations this is essential. It's like representing a smooth curve using a series of short straight lines.
In practice, to plot the marginal density \( f_{X}(t) \) discretely, you choose points within the defined range and create a histogram or line graph. Similarly, the marginal distribution function, \( F_{X}(t) \), is plotted point by point, by integrating the marginal density up to each chosen value of \( t \).

• This approach sacrifices perfect accuracy for a simple and understandable representation that still captures the essential behavior of the function.

Integration of Probability Density Functions

Integration is the mathematical tool that ties together many concepts in probability. When dealing with PDFs, integration allows you to compute probabilities as areas under a curve.
For continuous random variables, we use integration to move from the density function, which tells us the likelihood of a variable falling within an infinitesimal range, to practical probabilities over finite intervals.

• With our marginal density functions for X and Y, we're able to determine the probabilities for specific ranges of these variables by setting up and evaluating integrals over those ranges.
• For joint PDFs, the concept extends to double integrals, where we calculate the probability that X and Y fall inside a certain region by integrating the joint PDF over that region.