Question

Suppose the random vector \((\mathrm{X}, \mathrm{Y})\) is distributed with probability density, \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}+\mathrm{y}\) \(0<\mathrm{x}<1\) and \(=0 \quad 0<\mathrm{y}<1\) otherwise. Find \(E[X Y], E[X+Y]\) and \(E(X)\).

Short answer

The short answer is: E[XY] = \(\frac{7}{12}\) E[X+Y] = \(\frac{5}{3}\) E[X] = \(\frac{5}{6}\)

Step-by-step solution

Compute the Marginal Probability Density Functions of X and Y

To find the marginal probability density functions of X and Y, we will integrate f(x,y) with respect to the other variable. For the marginal probability density function of X, we integrate f(x,y) with respect to y: \(f_X(x) = \int_{0}^{1} (x + y) dy\) For the marginal probability density function of Y, we integrate f(x,y) with respect to x: \(f_Y(y) = \int_{0}^{1} (x + y) dx\)

Evaluate the Integrals to Find the Marginal Density Functions of X and Y

Now let's compute both the integrals: \(f_X(x) = \int_{0}^{1} (x + y) dy = \left[ xy + \frac{1}{2}y^2 \right]_0^1 = x + \frac{1}{2}\) \(f_Y(y) = \int_{0}^{1} (x + y) dx = \left[ \frac{1}{2}x^2 + xy \right]_0^1 = \frac{1}{2} + y\)

Compute E[XY], E[X+Y], and E[X] Using the Marginal Density Functions

Now we will use the marginal density functions to compute the expected values: \(E[XY] = \int_{0}^{1} \int_{0}^{1} xy(x + y) dx dy\) \(E[X+Y] = \int_{0}^{1} \int_{0}^{1} (x+y)(x + y) dx dy\) \(E[X] = \int_{0}^{1} x f_X(x) dx\)

Evaluate the Integrals to Find the Expected Values

Finally, we compute the integrals to find the expected values: \(E[XY] = \int_{0}^{1} \int_{0}^{1} xy(x + y) dx dy = \frac{7}{12}\) \(E[X+Y] = \int_{0}^{1} \int_{0}^{1} (x+y)(x + y) dx dy = \frac{5}{3}\) \(E[X] = \int_{0}^{1} x (x + \frac{1}{2}) dx = \frac{5}{6}\) Hence, the expected values are: E[XY] = \(\frac{7}{12}\) E[X+Y] = \(\frac{5}{3}\) E[X] = \(\frac{5}{6}\)

Key concepts

Random Variables

Random variables are fundamental elements in probability theory, used to model numerical outcomes of random processes. They are often denoted by capital letters like \(X\) or \(Y\).
In this article, \(X\) and \(Y\) are random variables related through a joint probability density function (pdf) \(f(x, y)\). This function describes the likelihood of \(X\) and \(Y\) taking particular values within a certain range.
For our specific problem, \(f(x, y) = x + y\) within the range of \(0 < x, y < 1\), providing a structured way to determine how probable certain outcomes are.
It's crucial to understand that while random variables can take on a range of values, the pdf gives a snapshot of their behavior over a defined interval. It's a key tool for calculating probabilities and expected outcomes.

Marginal Probability Density Function

Marginal probability density functions (pdfs) allow us to focus on individual random variables from a joint distribution by integrating over the unrelated variable.
To find the marginal pdf of \(X\), denoted \(f_X(x)\), we integrate the joint pdf \(f(x, y)\) with respect to \(y\) over its entire range. Similarly, for the marginal pdf of \(Y\) (\(f_Y(y)\)), we integrate with respect to \(x\).
Here's how it's done:

• \(f_X(x) = \int_{0}^{1} (x + y) \, dy\)
• \(f_Y(y) = \int_{0}^{1} (x + y) \, dx\)

This process strips down the joint distribution to focus on a single variable, summarizing its behavior independently of the other. Marginal pdfs are essential for deriving other statistical measures like expected values.

Expected Value

The expected value, or mean, gives the average outcome of a random variable if an experiment is repeated many times.
For a continuous random variable, it's found using the integral of the variable multiplied by its pdf.
For our problem, expected values are found as follows:

• \(E[XY]\) uses the joint pdf, evaluating \(E[XY] = \int_{0}^{1} \int_{0}^{1} xy(x + y) \, dx \, dy\)
• \(E[X+Y]\) evaluates the sum of the variables: \(E[X+Y] = \int_{0}^{1} \int_{0}^{1} (x+y)(x + y) \, dx \, dy\)
• \(E[X]\) uses the marginal pdf of \(X\): \(E[X] = \int_{0}^{1} x \, f_X(x) \, dx\)

The expected value helps us find the central tendency of our distribution, providing insights into what values the random variables are likely to take.

Integral Calculus

Integral calculus is a mathematical technique used to calculate areas under curves, which is critical in probability for finding probabilities and expected values.
When working with probability density functions, integration helps determine:

• The total probability over a given interval
• Marginal pdfs by integrating over the irrelevant variables
• Expected values by integrating the product of the variable and its pdf

For example, finding \(f_X(x)\) and the expected value involves integrating the relevant functions:
\( \int_{0}^{1} (x+y) \, dy \) or \( \int_{0}^{1} x \, f_X(x) \, dx \).
Mastering integration allows us to transform complex relationships in probability theory into calculated probabilities and expected outcomes, making predictions about random processes.