Problem 15 - 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. \(f_{X Y}(t, u)=\frac{3}{88}\left(2 t+3 u^{2}\right)\) for \(0 \leq t \leq 2,0 \leq u \leq 1+t\) \(F_{X Y}(1,1), P(X \leq 1, Y>1), P(|X-Y|<1)\)

Step-by-Step Solution

Answer

Follow the integration steps to find the marginal densities and cumulative probabilities. Use the regions defined by the conditions to set proper limits for integration and discrete approximation.

Step 1 - Sketch the Region of Definition

To sketch the region of definition for the joint probability density function (pdf) given by \(f_{XY}(t, u)\), plot the area bounded by the conditions \(0 \leq t \leq 2\) and \(0 \leq u \leq 1+t\), which forms a trapezoidal shape in the t-u plane.

Step 2 - Determine the Marginal Density \(f_X(t)\)

Integrate the joint pdf \(f_{XY}(t, u)\) over the variable \(u\) from 0 to \(1 + t\) to get the marginal density function for \(X\): \[f_X(t) = \int_0^{1+t} \frac{3}{88}(2t + 3u^2) \, du\].

Step 3 - Calculate the Marginal Density \(f_X(t)\)

Perform the integration process from Step 2. The result will be a function of \(t\) alone, representing the marginal density function for \(X\).

Step 4 - Determine the Marginal Density \(f_Y(u)\)

Integrate the joint pdf \(f_{XY}(t, u)\) over the variable \(t\) from 0 to 2 to get the marginal density function for \(Y\). For values of \(u\) greater than 1, the upper limit of integration will not be 2, but it will be reduced by the condition \(t \geq u-1\): \[f_Y(u) = \begin{cases} \int_0^{2} \frac{3}{88}(2t + 3u^2) \, dt, & \text{for } 0 \leq u \leq 1 \ \int_{u-1}^{2} \frac{3}{88}(2t + 3u^2) \, dt, & \text{for } 1 < u \leq 2 \end{cases}\].

Step 5 - Calculate the Marginal Density \(f_Y(u)\)

Perform the integration process from Step 4. The result will be a function of \(u\) alone, representing the marginal density function for \(Y\).

Step 6 - Discrete Approximation and Plot of \(f_X(t)\)

Choose a suitable number of equal intervals over the range \(0 \leq t \leq 2\), and evaluate \(f_X(t)\) at the midpoint of each interval. Create a bar graph where the height of each bar represents the result of \(f_X(t)\) on that interval.

Step 7 - Discrete Approximation and Plot of the Cumulative Distribution Function \(F_X(t)\)

Using the same intervals as in Step 6, add up the discrete approximations of \(f_X(t)\) from \(0\) to each point \(t\), representing the cumulative probability. Plot these cumulative values to approximate the marginal distribution function \(F_X(t)\).

Step 8 - Calculate Analytically \(F_{XY}(1,1)\)

Integrate the joint pdf \(f_{XY}\) over the region from \(0\) to \(1\) for \(t\) and from \(0\) to \(1\) for \(u\) to find the joint cumulative distribution function at point \((1, 1)\): \[F_{XY}(1, 1) = \int_0^1 \int_0^1 \frac{3}{88}(2t + 3u^2) \, du \, dt\].

Step 9 - Calculate Analytically \(P(X \leq 1, Y>1)\)

Integrate the joint pdf \(f_{XY}\) over the region defined by \(t\) from \(0\) to \(1\), and \(u\) from \(1\) to \(1 + t\): \[P(X \leq 1, Y > 1) = \int_0^1 \int_1^{1+t} \frac{3}{88}(2t + 3u^2) \, du \, dt\].

Step 10 - Calculate Analytically \(P(|X-Y|<1)\)

The region where \(|X - Y| < 1\) consists of two integrals due to the absolute value condition. Integrate the joint pdf \(f_{XY}\) within the region that satisfies this condition: \[P(|X-Y|<1) = \int_{region} \frac{3}{88}(2t + 3u^2) \, du \, dt\].

Step 11 - Determine Discrete Approximation of the Indicated Probabilities

Using the same intervals as in Step 6 and Step 7, approximate the probabilities for \(F_{XY}(1,1)\), \(P(X \leq 1, Y>1)\), and \(P(|X-Y|<1)\) by summing up the discrete values of the joint pdf within the regions corresponding to these probability conditions.

Key Concepts

Joint Probability Density Function

Understanding the joint probability density function (pdf) is critical for grasping the fundamentals of multivariate probability. It essentially describes how probability is distributed over two random variables, X and Y. As in the provided exercise, the function is given by \(f_{XY}(t, u)\), which specifies the likelihood of the random variables falling within a certain range.

To visualize the domain of this function, we can sketch the region of definition. Here, we have the conditions \(0 \leq t \leq 2\) and \(0 \leq u \leq 1+t\), which determine the possible values that t and u can take. The resulting graph is a trapezoidal shape, showcasing that the probability is non-zero only within this region.

Within this context, integrating the joint pdf over one of the variables—while keeping the other constant—yields the marginal density function of the remaining variable. This method of integration is crucial as it boils down a joint distribution into a univariate distribution, offering a simpler perspective on the probability distribution of a single variable while accounting for the presence of another.

Cumulative Distribution Function

The cumulative distribution function (CDF), represented as \(F_X(t)\) or \(F_{XY}(t, u)\), plays a pivotal role in probability and statistics by describing the probability that a random variable takes a value less than or equal to a specific number. For a joint pdf, the CDF \(F_{XY}(t, u)\) provides the probability that variable X is less than or equal to t and variable Y is less than or equal to u simultaneously.

The CDF is found by integrating the joint pdf over all values up to the specified limits. In our exercise example, to find \(F_{XY}(1,1)\), we integrate \(f_{XY}(t, u)\) from 0 to 1 for both t and u. This integral gives us the probability lying within the lower left corner of our defined region, up to the point (1,1).

The cumulative function is especially helpful when we need to calculate probabilities over continuous intervals or when we seek to understand the probability distribution up to a certain threshold. Incrementally integrating or summing the density function leads to the cumulative probability, displaying how the likelihood accumulates as we move across the domain of our random variables.

Integration in Probability

Integration is a fundamental tool in probability theory often used to compute marginal densities, cumulative distribution functions, and various probabilities involving continuous random variables. In essence, integration sums up infinitely many infinitesimal probabilities across a range of values to give a finite probability measure.

The exercise presents several instances where integration is key. For instance, calculating the marginal density functions \(f_X(t)\) and \(f_Y(u)\) or the probability \(P(|X-Y|<1)\) involves setting up and evaluating integrals of the joint pdf over specific regions. The regions over which we integrate are defined by the conditions given in the problem, such as the trapezoidal region for the joint pdf in this particular exercise.

Integration in probability can sometimes become complex, especially with conditional or intricate boundaries for the variables involved. This complexity is evident when we aim to calculate probabilities like \(P(X \leq 1, Y > 1)\), where integration boundaries depend on the conditions placed on the variables. Nevertheless, by carefully setting up the integrals and considering the boundaries, we can use integration to extract meaningful information from the probability density functions.