Problem 13 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Remember that we found on page 95 that the expected area of a building was \(33 \frac{1}{3} \mathrm{~m}^{2}\), whereas the square of the expected width was only \(25 \mathrm{~m}^{2}\). This phenomenon is more general: show that for any random variable \(X\) one has \(\mathrm{E}\left[X^{2}\right] \geq(\mathrm{E}[X])^{2} .\)

Step-by-Step Solution

Answer

For any random variable \(X\), \(E[X^2] \geq (E[X])^2\) due to non-negative variance.

Step 1: Understand the Problem

We need to prove that for any random variable \(X\), the expected value of its square \(E[X^2]\) is greater than or equal to the square of its expected value \((E[X])^2\). This inequality is known as the variance inequality, part of Jensen's Inequality.

Step 2: Recall the Definition of Variance

Variance of a random variable \(X\), denoted as \(Var(X)\), is defined as \(Var(X) = E[X^2] - (E[X])^2\). This can be reorganized to show the inequality \(E[X^2] = Var(X) + (E[X])^2\).

Step 3: Analyze Variance

Since the variance of a random variable, \(Var(X)\), represents the spread of the distribution of \(X\) and is always non-negative (\(Var(X) \geq 0\)), we can conclude that \(E[X^2] \geq (E[X])^2\).

Step 4: Formalize the Conclusion

By knowing that \(Var(X) \geq 0\), we apply this to the formula derived in step 2: \(E[X^2] = Var(X) + (E[X])^2\). Hence, \(E[X^2] \geq (E[X])^2\) is confirmed as \(Var(X)\) is non-negative.

Key Concepts

Random Variables

Think of a random variable as a bridge between abstract probability theory and the real world filled with uncertainties. It is a concept that helps to assign numerical values to random events, allowing us to work with them systematically. Consider a random variable as a function that gives each outcome of a random process a number. For instance, if you were to roll a die, the value shown on the die is a random variable, which can be 1 through 6. Random variables are classified into two main types:

• Discrete Random Variables: These variables can take on a countable number of distinct values. An example is rolling a die, where the variable can be 1, 2, 3, 4, 5, or 6.
• Continuous Random Variables: These variables can take any value within a given range. For instance, the exact temperature at any moment can be any value within a range and is not countable.

Understanding random variables is crucial in probability theory because they allow us to model real-world situations mathematically, facilitating analysis and decision-making.

Jensen's Inequality

Jensen's Inequality is a fundamental result in probability and optimization theory, stating that for any convex function, the function's value at the expectation of a random variable is less than or equal to the expectation of the function applied to the random variable. Mathematically, if \( ext{f} \) is a convex function and \( ext{X} \) is a random variable, then:\[ \mathrm{E}[\text{f}(X)] \geq \text{f}(\mathrm{E}[X]) \]This inequality shows up in various contexts, such as economics, information theory, and operational research, where it describes how averaging affects convex and concave functions.In our context, when we examine \( E[X^2] \geq (E[X])^2 \), we see Jensen's Inequality at play. Since the square function \( \text{f}(x) = x^2 \) is convex, it leads us to infer that the expected value of \( X^2 \) is always at least as large as the square of its expected value. This property of convex functions makes Jensen’s Inequality an incredibly useful tool for proving various inequalities and is essential for understanding the behavior of averages.

Variance

Variance is a statistical concept that gives us deep insight into the spread and reliability of our data. It measures how much the values of a random variable deviate from their mean, providing a quantitative way to express the variable's volatility.Let's break down the concept:- **Definition:** Variance of a random variable \( X \), denoted as \( ext{Var}(X) \), is calculated as the expected value of the squared deviation of \( X \) from its mean: \[ \text{Var}(X) = \mathrm{E}[(X - ext{E}[X])^2] \].- **Formula:** This formula can be re-arranged to: \[ \text{Var}(X) = E[X^2] - (E[X])^2 \] As such, variance is the difference between the expected value of the square of \( X \) and the square of its expected value. Because variance is always non-negative (since it is an expectation of a squared quantity), we derive the inequality \( E[X^2] \geq (E[X])^2 \), confirming our earlier analysis. Variance is a crucial parameter because it not only informs us about the `average' value of a dataset but also about the `spread' of these values, making it instrumental in fields such as finance, engineering, and any discipline that deals with uncertainty or variability.