Problem 40 - A First Course in Probability

Question

The joint density function of \(X\) and \(Y\) is given by $$f(x, y)=\frac{1}{y} e^{-(y+x / y)}, \quad x>0, y>0$$ Find \(E[X], E[Y],\) and show that \(\operatorname{Cov}(X, Y)=1\)

Step-by-Step Solution

Answer

We find the marginal densities of \(X\) and \(Y\), and then use them to compute the expectations \(E[X]\) and \(E[Y]\). Lastly, we calculate the covariance \(\operatorname{Cov}(X, Y)\) using the expectations we obtained. We find that \(\operatorname{Cov}(X, Y) = 1\).

Step 1: Find the marginal densities of X and Y

To find the marginal densities, we must integrate the joint density function \(f(x, y)\) with respect to the other variable. The marginal probability density function of X: \[f_X(x) = \int_{0}^{\infty} f(x, y) dy = \int_{0}^{\infty} \frac{1}{y} e^{-(y + x/y)} dy\] The marginal probability density function of Y: \[f_Y(y) = \int_{0}^{\infty} f(x, y) dx = \int_{0}^{\infty} \frac{1}{y} e^{-(y + x/y)} dx\]

Step 2: Calculate E[X] and E[Y] using the marginal densities

Now that we have the marginal densities, we can use them to find \(E[X]\) and \(E[Y]\). Expected value of X: \[E[X] = \int_{0}^{\infty} x f_X(x) dx \] Expected value of Y: \[E[Y] = \int_{0}^{\infty} y f_Y(y) dy \]

Step 3: Compute the covariance of X and Y

Lastly, we will compute the covariance of \(X\) and \(Y\). The covariance is given by the following formula: \[\operatorname{Cov}(X,Y) = E[XY] - E[X]E[Y]\] First, we'll compute the expected value of \(XY\), \(E[XY]\): \[E[XY] = \int_{0}^{\infty} \int_{0}^{\infty} xy f(x,y) dx dy\] After finding \(E[XY]\) and using the expected values \(E[X]\) and \(E[Y]\) computed in step 2, we'll plug these values into the covariance formula: \[\operatorname{Cov}(X,Y) = E[XY] - E[X]E[Y]\] If our calculation yields \(\operatorname{Cov}(X, Y) = 1\), we will have successfully shown that the covariance of \(X\) and \(Y\) is 1.

Key Concepts

Marginal Densities

Understanding marginal densities is crucial when analyzing joint probability density functions. Marginal density functions describe the probability distribution of a subset of the variables within a multivariate distribution, irrespective of the other variables.

In the given exercise, we encounter a joint density function of two variables, X and Y. To comprehend the probability distribution of X independently of Y, and vice versa, we must find their marginal densities, denoted as \(f_X(x)\) for X and \(f_Y(y)\) for Y. This is achieved by integrating the joint density function over the entire range of the other variable.

The integration process effectively 'sums out' the influence of the variable not being considered, leaving us with a density function solely for the variable of interest. This is foundational for calculating individual expected values and variances needed for further analysis, such as determining probabilities of certain events or identifying relationships between the variables.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency, indicating where the majority of the values in its probability distribution tend to occur. For continuous random variables, the expected value is calculated by integrating the product of the variable's value and its probability density function over all possible values.

In practice, we multiply each potential outcome by its probability and sum these products to find the expected value. In the context of marginal densities found from joint probability density functions, we take the marginal density function for each variable \(f_X(x)\) and \(f_Y(y)\), multiply by the variable itself (e.g., x or y), and integrate over all possible values to find \(E[X]\) and \(E[Y]\), respectively.

Covariance Calculation

Covariance provides a measure of how much two random variables change together and is essential in understanding their relationship. A positive covariance signifies that the variables tend to increase or decrease together, while a negative covariance indicates that one variable tends to increase when the other decreases.

The calculation of covariance involves finding the expected value of the product of the variables' deviations from their respective means. In formal terms, \(\operatorname{Cov}(X, Y)\) is the expected value of \((X - E[X])(Y - E[Y])\). However, it is often computed using the formula \(\operatorname{Cov}(X,Y) = E[XY] - E[X]E[Y]\), which requires calculating the expected value of the product of X and Y, as well as the expected values of X and Y individually, as seen in the mentioned exercise.

Density Function Integration

Integration over a density function is the method used to calculate probabilities, expected values, variances, and covariances for continuous random variables. When we integrate the joint probability density function over one variable, we are obtaining the marginal density for the other variable. Likewise, integrating the product of a variable and its marginal density gives us the expected value of that variable.

The process can be challenging if the integrals are complex, which is why knowing and applying integration techniques, such as substitution or integration by parts, is invaluable. As shown in the exercise, correctly setting up the integrals based on the joint density function and marginal densities is the stepping stone to solving for properties of the probability distribution, such as the covariance.