Problem 214 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(f_{X, Y}(x, y)=\frac{2}{2}\left(x^{2}+y^{2}\right), 0 \leq x \leq 1\), \(0 \leq y \leq 1\). Find \(\operatorname{Var}(X+Y)\).

Step-by-Step Solution

Answer

The variance of \(X+Y\) is computed by adding the individual variances and twice the covariance.

Step 1: Compute Expectations \(E[X]\) and \(E[Y]\)

First we compute the expectations of \(X\) and \(Y\). The expectation of a random variable \(\text{X}\) is the sum of the product of each outcome and its likelihood, and can also be interpreted as a 'long-run average'. Therefore, \(E[X] = \int_0^1 x f_{X, Y}(x, y) dx = \int_0^1 x (x^{2}+y^{2}) dx\) and similarly \(E[Y] = \int_0^1 y f_{X, Y}(x, y) dy = \int_0^1 y (x^{2}+y^{2}) dy\).

Step 2: Compute Second Moment \(E[X^2]\) and \(E[Y^2]\)

Next, compute the second moments of \(X\) and \(Y\). In a similar way, \(E[X^2] = \int_0^1 (x^2) f_{X, Y}(x, y) dx = \int_0^1 (x^2) (x^{2}+y^{2}) dx\) and \(E[Y^2] = \int_0^1 (y^2) f_{X, Y}(x, y) dy = \int_0^1 (y^2) (x^{2}+y^{2}) dy\).

Step 3: Compute Variances \(Var[X]\) and \(Var[Y]\)

Once we have the expectations and second moments of \(X\) and \(Y\), we can compute their variances using the formula \(Var[X]=E[X^2]-(E[X])^2\) and \(Var[Y]=E[Y^2]-(E[Y])^2\).

Step 4: Compute Covariance \(Cov[X,Y]\)

The covariance between two variables \(X\) and \(Y\) measures how the two are linearly related. To calculate, \(Cov[X,Y] = E[XY] - E[X]E[Y]\). We can get \(E[X]\), \(E[Y]\), and \(E[XY]\) in a similar way as previous steps.

Step 5: Compute Variance \(Var[X+Y]\)

Finally, substitute the values of \(Var[X]\), \(Var[Y]\), and \(Cov[X,Y]\) into the formula \(Var[X+Y]=Var[X]+Var[Y]+2Cov[X,Y]\) to get the variance of \(X+Y\).

Key Concepts

Joint Probability Density Function

A Joint Probability Density Function (joint PDF) describes the likelihood of two continuous random variables occurring together in a specific range. For instance, if we have two variables, X and Y, the joint PDF, denoted as \( f_{X,Y}(x,y) \), represents their combined probabilities. This function allows us to infer probabilities of events involving both X and Y simultaneously.

In our specific exercise, the joint PDF given is \( f_{X,Y}(x,y) = x^2 + y^2 \) for \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \). This means that both X and Y are limited to values between 0 and 1, and their joint probability is affected by the sum of their squared values. One key aspect of joint PDFs is that they must integrate to 1 over all possible values of \( x \) and \( y \), ensuring that the total probability is 100%.

• Joint PDFs are crucial for finding probabilities involving two or more variables.
• They help understand the interactions between random variables.
• Must satisfy the condition that the integral over the entire possible space equals 1.

Expectation and Expected Value

Expectation, often termed the expected value, is a fundamental concept for understanding the average or 'mean' outcome of a random variable over numerous trials.

For a single random variable, the expectation \( E[X] \) is calculated as the integral of \( x \) times its probability density function, \( f(x) \), over its range. In practice, it is the sum of all possible values that the variable can take, each weighted by its probability.

In exercises like ours, where a joint PDF is involved, we calculate expected values for each variable separately, \( E[X] \) and \( E[Y] \), using:

• \( E[X] = \int x \, f_{X,Y}(x,y) \, dx \)
• \( E[Y] = \int y \, f_{X,Y}(x,y) \, dy \)

These computations provide the average outcomes we expect from X and Y, providing insights into their behavior in the context of the whole joint distribution.

Covariance of Random Variables

Covariance is a measure that signals how two random variables change together. If they increase or decrease in tandem, their covariance is positive. Conversely, if one increases while the other decreases, their covariance is negative.

Mathematically, covariance between two variables, X and Y, is determined by the formula:

• \( Cov[X, Y] = E[XY] - E[X]E[Y] \).

By computing \( E[XY] \), the expected value of their product, and subtracting the product of their individual expectations, \( E[X]E[Y] \), we get an idea of their mutual dependency.

In problems regarding variance, such as finding \( Var(X+Y) \), covariance is crucial as it accounts for the interaction between X and Y. It directly influences the result via the formula:

• \( Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y) \).

This inclusion effectively adjusts for how the variables are correlated, providing a more comprehensive understanding of their total variability.