Problem 31 - Calculus for Biology and Medicine

Question

We have two formulas for computing the variance of \(X\), namely, $$ \operatorname{var}(X)=E\left[(X-E(X))^{2}\right] $$ and $$ \operatorname{var}(X)=E\left(X^{2}\right)-[E(X)]^{2} $$ (a) Explain why \(\operatorname{var}(X) \geq 0\). (b) Use your results in (a) to explain why $$ E\left(X^{2}\right) \geq[E(X)]^{2} $$

Step-by-Step Solution

Answer

The variance is non-negative, which implies \( E(X^2) \geq [E(X)]^2 \).

Step 1: Understanding Variance

Variance of a random variable \(X\), \( \operatorname{var}(X) \), measures the spread or dispersion of the values \(X\) takes around its expected value \(E(X)\). The variance quantifies the average squared deviations from the mean. Therefore, it must be a non-negative number, as it represents a squared quantity (distance), which cannot be negative.

Step 2: Deriving Variance Formula

The formula \( \operatorname{var}(X)=E[(X-E(X))^2] \) shows us that variance is the expectation of the squared deviation of \(X\) from its mean. Since this is a squared value, it must be \( \geq 0 \). Thus, \( \operatorname{var}(X) \geq 0 \).

Step 3: Connecting Second Formula of Variance

Using the other formula \( \operatorname{var}(X)=E(X^2) - [E(X)]^2 \), since we know \( \operatorname{var}(X) \geq 0 \), it follows that \( E(X^2) - [E(X)]^2 \geq 0 \). Rearranging the terms gives us \( E(X^2) \geq [E(X)]^2 \).

Step 4: Conclusion of the Comparison

From the non-negativity of the variance, we directly show the inequality \( E(X^2) \geq [E(X)]^2 \). This means that the expected value of the square of \(X\) is always at least as large as the square of the expected value of \(X\), verifying typical relationships in probability theory.

Key Concepts

Random Variable

A random variable is a fundamental concept in probability theory that assigns a numerical value to each outcome in a sample space. In simple terms, it helps us quantify the outcomes of a random process. Random variables can be discrete, where they take specific values like rolling a dice, or continuous, where they can take any value within a range, such as measuring the temperature on a particular day.

Random variables are denoted by capital letters like \(X\), \(Y\), etc. When we discuss properties such as variance or expected value, we are usually talking about these random variables. By helping us focus on the numerical outcomes of random events, random variables allow us to apply mathematical tools to analyze and draw conclusions about probability distributions.

Expected Value

Expected value is a critical concept in probability that represents the average or mean value of a random variable over many trials or experiments. It's the center point or the balance of the probability distribution.

For a discrete random variable \(X\), the expected value \(E(X)\) is calculated by multiplying each possible outcome by its probability and summing them up. For continuous random variables, you integrate over the possible values.

In mathematical terms, the expected value is given by:

• Discrete: \(E(X) = \sum x_i \cdot P(x_i)\)
• Continuous: \(E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx\)

The expected value gives us a single number that describes the average outcome of \(X\) when the experiment is repeated many times.

Probability Theory

Probability theory is the branch of mathematics that deals with the analysis of random phenomena. It provides a framework for describing the likelihood of various outcomes in any situation involving uncertainty. This theory introduces the concept of probability, which quantifies how likely an event is possibly to occur, on a scale from 0 to 1.

Applying probability theory allows us to create mathematical models for real-world situations. It encompasses a vast array of topics including, but not limited to, random variables, expected value, and variance. Through probability distribution functions, one can determine how probabilities are assigned across different possible outcomes of a random variable. By understanding these principles, we can make informed predictions and decisions even in the face of randomness.

One of the key points of probability theory is understanding how concepts like expected value and variance interact to give us meaningful insight into the behavior of systems. For instance, understanding why the expected value of a square is greater than or equal to the square of the expected value helps when analyzing and comparing different variance outcomes.