Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that the random variables \((X, Y)\) have joint \(\mathrm{PDF}\) \(f(x, y)=\left\\{\begin{array}{ll}\frac{3}{256}\left(x^{2}+y^{2}\right), & \text { if } 0 \leq x \leq y ; 0 \leq y \leq 4 \\ 0, & \text { otherwise }\end{array}\right.\) Find each of the following: (a) \(P(X>2)\) (b) \(P(X+Y \leq 4)\) (c) \(E(X+Y)\)

Step-by-Step Solution

Answer

Calculate multiple integrals for probabilities and expected values for given joint PDF.

Step 1: Understand the Problem

We have two random variables \(X\) and \(Y\) with a joint probability density function (PDF) defined within certain limits. We need to calculate the probabilities given in parts (a) and (b) and the expected value in part (c).

Step 2: Find P(X > 2)

We want the probability that \(X > 2\). The region where the PDF is non-zero is given by \(0 \leq x \leq y\) and \(0 \leq y \leq 4\). Therefore, the limits for \(X\) are \([2, y]\) and for \(Y\) are \([x, 4]\). Set up the double integral: \[P(X > 2) = \int_{2}^{4} \int_{x}^{4} \frac{3}{256}(x^2 + y^2) \; dy \; dx\]. Evaluate the inner integral with respect to \(y\), then the outer integral with respect to \(x\).

Step 3: Calculate Inner Integral for P(X > 2)

First, solve the inner integral \[\int_{x}^{4} \frac{3}{256}(x^2 + y^2) \; dy = \frac{3}{256} \left[x^2y + \frac{y^3}{3}\right]_{x}^{4}.\]Substitute the limits of integration for \(y\) and simplify.

Step 4: Calculate Outer Integral for P(X > 2)

Now solve the outer integral \[\int_{2}^{4} \left(\frac{3}{256} \left[x^24 + \frac{4^3}{3} - x^2x - \frac{x^3}{3}\right]\right) \; dx.\]Carry out the integration and simplify to find \(P(X > 2)\).

Step 5: Find P(X+Y ≤ 4)

For the probability \(P(X+Y \leq 4)\), note that the applicable region within the joint PDF bounds is bounded by the line \(x + y = 4\) and the axes within this region. Thus, set up the double integral \[P(X + Y \leq 4) = \int_{0}^{4} \int_{0}^{4-y} \frac{3}{256}(x^2 + y^2) \; dx \; dy\].Evaluate the inner integral and then the outer integral.

Step 6: Calculate Inner Integral for P(X+Y ≤ 4)

Evaluate the inner integral:\[\int_{0}^{4-y} \frac{3}{256}(x^2 + y^2) \; dx = \left[ \frac{3}{256} (\frac{x^3}{3} + y^2x) \right]_{0}^{4-y}\].This simplifies to \(\frac{3}{256} \left[ \frac{(4-y)^3}{3} + y^2(4-y) \right]\).

Step 7: Calculate Outer Integral for P(X+Y ≤ 4)

Now, evaluate the outer integral:\[\int_{0}^{4} \frac{3}{256} \left[ \frac{(4-y)^3}{3} + y^2(4-y) \right] dy\].Carry out the integration and simplify to find \(P(X+Y \leq 4)\).

Step 8: Determine E(X+Y)

For the expected value \(E(X+Y)\), calculate:\[E(X+Y)= \int_{0}^{4} \int_{0}^{y} (x+y) \frac{3}{256}(x^2+y^2) \; dx \; dy\]Split into two integrals, one for \(E(X)\) and one for \(E(Y)\), solve them and sum the results:\[ E(X) = \int_{0}^{4} \int_{0}^{y} x \frac{3}{256}(x^2+y^2) \; dx \; dy\]\[ E(Y) = \int_{0}^{4} \int_{0}^{y} y \frac{3}{256}(x^2+y^2) \; dx \; dy\].Solve each integral separately and add the outcomes.

Key Concepts

Random Variables

Random variables are essential components in probability and statistics. These variables represent possible outcomes of random phenomena. In the context of the given exercise, we have two random variables, \(X\) and \(Y\), which together are governed by a joint probability density function (PDF). This joint PDF is a mathematical function that provides the likelihood of \(X\) and \(Y\) occurring together within their specified range.

In continuous domains, random variables can take any real value, and the joint PDF helps us understand the probability of these variables being in certain intervals. Moreover, random variables can be dependent or independent, impacting how probabilities and expectations are calculated. For \(X\) and \(Y\) in this exercise, the joint PDF is non-zero only within the specified limits, meaning that outside this range, the probability of occurrence is zero.

Understanding random variables in the context of a joint PDF involves recognizing this function's role in providing a comprehensive picture of where \(X\) and \(Y\) are most likely to occur. To analyze \(P(X > 2)\), \(P(X+Y \leq 4)\), and the expected value, we essentially evaluate how these variables behave together, as determined by their joint distribution.

Probability Calculation

Probability calculations involving joint PDFs require setting up and evaluating integrals over specified ranges. This is crucial in determining the likelihood of events concerning the variables \(X\) and \(Y\).

To calculate \(P(X > 2)\), one has to consider the integral of the joint PDF over a region where \(X\) is greater than 2, yet respects the condition \(0 \leq x \leq y \leq 4\). This leads to a double integral, first over \(y\) and then over \(x\). By solving these integrals step-by-step and using the limits of integration, we determine the desired probability.

For \(P(X+Y \leq 4)\), the calculation involves a boundary formed by \(x+y = 4\). Again, a double integral approach is adopted. Here, the integration occurs within the limits defined by the PDF non-zero region and the said boundary. Each step in solving the integrals contributes to obtaining the final probability, using the properties of integration to handle variables raised to powers and fixed limits.

Integrals in probability provide the cumulative probability over a continuum, describing chances of occurrence within defined bounds. This way, understanding and calculating probabilities translate to determining the likelihood sums over these ranges, examined through integration rules and their evaluations.

Expected Value

The expected value, or expectation, of random variables \(X\) and \(Y\) gives us insights into the average or mean outcome we can anticipate from these variables over numerous trials. When dealing with joint PDFs, the expected value combines the effects of both variables across their entire distribution range.

In the exercise, we calculate \(E(X+Y)\), indicating the expected sum of \(X\) and \(Y\). To find this, you split the problem into more manageable parts: finding \(E(X)\) and \(E(Y)\) separately and then combining them. This involves setting up separate integrals where each variable is multiplied by the joint PDF within its respective integral bounds.

By integrating these expressions, we account for how likely each outcome is weighted by the contribution of \(x\) or \(y\), thus generating an expected value. Summing the results from these integrations gives the combined expectation.

The process reflects on both the inherent nature of each variable and their collective behavior, leading us to a comprehensive understanding of their joint impact. In probability theory, this concept is valuable for making predictions on the behavior of variables, optimizing decision-making, and formulating strategic responses based on average expected scenarios.