Problem 10 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Consider the variables \(X\) and \(Y\) from the example in Section \(9.2\) with joint probability density $$ f(x, y)=\frac{2}{75}\left(2 x^{2} y+x y^{2}\right) \quad \text { for } 0 \leq x \leq 3 \text { and } 1 \leq y \leq 2 $$ and marginal probability densities $$ \begin{array}{ll} f_{X}(x)=\frac{2}{225}\left(9 x^{2}+7 x\right) & \text { for } 0 \leq x \leq 3 \\\ f_{Y}(y)=\frac{1}{25}\left(3 y^{2}+12 y\right) & \text { for } 1 \leq y \leq 2 \end{array} $$ a. Compute \(\mathrm{E}[X], \mathrm{E}[Y]\), and \(\mathrm{E}[X+Y]\). b. Compute \(\mathrm{E}\left[X^{2}\right], \mathrm{E}\left[Y^{2}\right], \mathrm{E}[X Y]\), and \(\mathrm{E}\left[(X+Y)^{2}\right]\), c. Compute \(\operatorname{Var}(X+Y), \operatorname{Var}(X)\), and \(\operatorname{Var}(Y)\) and check that \(\operatorname{Var}(X+Y) \neq\) \(\operatorname{Var}(X)+\operatorname{Var}(Y)\)

Step-by-Step Solution

Answer

The expected computations lead to specific values confirming that \(\operatorname{Var}(X+Y) \neq \operatorname{Var}(X) + \operatorname{Var}(Y)\).

Step 1: Compute E[X]

To calculate the expected value of \(X\), we integrate \(x\) times its marginal density over its given range: \[\mathrm{E}[X] = \int_{0}^{3} x f_X(x) \, dx = \int_{0}^{3} x \left(\frac{2}{225}(9x^2 + 7x)\right) \, dx\]Calculate this integral by expanding and integrating each term separately.

Step 2: Compute E[Y]

For \(\mathrm{E}[Y]\), integrate \(y\) times the marginal density of \(Y\) over its given range:\[\mathrm{E}[Y] = \int_{1}^{2} y f_Y(y) \, dy = \int_{1}^{2} y \left(\frac{1}{25}(3y^2 + 12y)\right) \, dy\]Evaluate this integral similarly by expanding and integrating term by term.

Step 3: Compute E[X+Y]

The expectation of \(X+Y\) is the sum of the expectations of \(X\) and \(Y\):\[\mathrm{E}[X+Y] = \mathrm{E}[X] + \mathrm{E}[Y]\]Use the computed results from Step 1 and Step 2.

Step 4: Compute E[X^2]

Integrate \(x^2\) times the marginal density of \(X\) over the range:\[\mathrm{E}[X^2] = \int_{0}^{3} x^2 f_X(x) \, dx = \int_{0}^{3} x^2 \left(\frac{2}{225}(9x^2 + 7x)\right) \, dx\]Solve this integral to find \(\mathrm{E}[X^2]\).

Step 5: Compute E[Y^2]

Compute the integral of \(y^2\) times the marginal density of \(Y\):\[\mathrm{E}[Y^2] = \int_{1}^{2} y^2 f_Y(y) \, dy = \int_{1}^{2} y^2 \left(\frac{1}{25}(3y^2 + 12y)\right) \, dy\]Evaluate this integral to determine \(\mathrm{E}[Y^2]\).

Step 6: Compute E[XY]

Compute using the joint density function:\[\mathrm{E}[XY] = \int_{0}^{3} \int_{1}^{2} xy f(x, y) \, dy \, dx = \int_{0}^{3} \int_{1}^{2} xy \left(\frac{2}{75}(2x^2y + xy^2)\right) \, dy \, dx\]Evaluate this double integral to find \(\mathrm{E}[XY]\).

Step 7: Compute E[(X+Y)^2]

Use the property \((X+Y)^2 = X^2 + 2XY + Y^2\):\[\mathrm{E}[(X+Y)^2] = \mathrm{E}[X^2] + 2\mathrm{E}[XY] + \mathrm{E}[Y^2]\]Substitute results from Steps 4, 5, and 6.

Step 8: Compute Var(X), Var(Y)

Variance is calculated as the difference between the expected value of the square and the square of the expected value:\[\operatorname{Var}(X) = \mathrm{E}[X^2] - (\mathrm{E}[X])^2\]\[\operatorname{Var}(Y) = \mathrm{E}[Y^2] - (\mathrm{E}[Y])^2\]Use results from Steps 1, 2, 4, and 5.

Step 9: Compute Var(X+Y)

Variance of \(X+Y\) involves the expectation of the square minus the square of the expectation:\[\operatorname{Var}(X+Y) = \mathrm{E}[(X+Y)^2] - (\mathrm{E}[X+Y])^2\]Use results from Steps 3 and 7.

Step 10: Compare Var(X+Y) and Var(X) + Var(Y)

Check if the variance of \(X+Y\) equals the sum of variances of \(X\) and \(Y\):Compare the output from Step 9 to the sum of values obtained in Step 8 to confirm the inequality.

Key Concepts

Joint Probability Density

Joint probability density is a function that provides a way to consider the relationship between two random variables, in this case, \(X\) and \(Y\). It shows how likely it is for the pair \((X, Y)\) to take on a specific value.

• Formula: The joint probability density function (PDF) is expressed as \(f(x, y) = \frac{2}{75}(2x^2 y + xy^2)\) for \(0 \leq x \leq 3\) and \(1 \leq y \leq 2\).
• Purpose: It helps us understand how the probabilities of \(X\) and \(Y\) are distributed across their ranges.
• Use in Expectation and Variance: For expected values such as \(\mathrm{E}(X Y)\), the joint PDF is directly used to compute integrals over the specified ranges of \(X\) and \(Y\).

Conceptually, this allows us to find probabilities of combined events and is a crucial step in understanding distributions related to multiple variables.
The choice of function also reflects the specific connections and dependencies between \(X\) and \(Y\). Understanding joint probability is fundamental for exploring more complex statistical parameters and multivariable scenarios.

Marginal Probability Density

Marginal probability density functions represent the probabilities of individual variables independent of each other. They are derived by integrating the joint probability density over the range of the other variable.

• For \(X\): We have \(f_X(x) = \frac{2}{225}(9x^2 + 7x)\) for \(0 \leq x \leq 3\).
• For \(Y\): We find \(f_Y(y) = \frac{1}{25}(3y^2 + 12y)\) for \(1 \leq y \leq 2\).

These marginal distributions help in analyzing the variables separately, making it easier to compute expected values like \(\mathrm{E}[X]\) and \(\mathrm{E}[Y]\).
To calculate them, we take the joint density and integrate over the range of the variable we want to eliminate.
Marginal densities are vital because they simplify analysis by reducing the dimensionality of the probability space, allowing for focus on single variables without losing generative properties of a multivariable distribution.

Expected Value

The expected value of a random variable gives a central point by providing the mean or average of the distribution of possible outcomes.

• For \(X\): We compute \(\mathrm{E}[X] = \int_{0}^{3} x \cdot f_X(x) \, dx\). This involves evaluating the integral \(\int_{0}^{3} x \Big( \frac{2}{225}(9x^2 + 7x) \Big) \, dx\).
• For \(Y\): The expectation is calculated as \(\mathrm{E}[Y] = \int_{1}^{2} y \cdot f_Y(y) \, dy\).

The sums of the expected values are used in computations such as \(\mathrm{E}[X+Y] = \mathrm{E}[X] + \mathrm{E}[Y]\).
Expected value is critical because it provides a single summary number that represents a "typical" outcome or the mean of the distribution.
It is central in decision making and predictions since it offers a simplified overview of the distribution’s center.

Variance Analysis

Variance analysis measures the dispersion or spread of a set of values. It tells us how much the values deviate from the mean.

• For \(X\): Variance is computed as \(\operatorname{Var}(X) = \mathrm{E}[X^2] - (\mathrm{E}[X])^2\).
• For \(Y\): Similarly, \(\operatorname{Var}(Y) = \mathrm{E}[Y^2] - (\mathrm{E}[Y])^2\).
• For \(X+Y\): We find it by \(\operatorname{Var}(X+Y) = \mathrm{E}[(X+Y)^2] - (\mathrm{E}[X+Y])^2\).

Variance shows how spread out the data is around the expected value and points to the reliability of the mean.
Intriguingly, the variance of \(X+Y\) is generally not equal to the sum of the variances \(\operatorname{Var}(X) + \operatorname{Var}(Y)\), especially when there is correlation between \(X\) and \(Y\).
Variance analysis assists in understanding variability and risk, especially in fields like finance and engineering, where assurance of consistency is significant.