Question

Define a random variable \(X\) on the sample space \(\Omega\) by setting \(X(\omega)=3\) for all \(\omega \in \Omega\). What is \(E(X) ? \operatorname{Var}(X) ?\)

Short answer

\(E(X) = 3\), \(\operatorname{Var}(X) = 0\).

Step-by-step solution

Understand the Random Variable

The random variable \(X\) is defined such that it assigns the value \(3\) to every outcome \(\omega\) in the sample space \(\Omega\). This means that \(X(\omega) = 3\) for all \(\omega\).

Determine the Expectation \(E(X)\)

The expectation of a random variable \(X\), denoted \(E(X)\), is the weighted average of all possible values of \(X\) with respect to their probabilities. Since \(X = 3\) with probability 1 (because every outcome \(\omega\) maps to 3), we have \(E(X) = 3 \times 1 = 3\).

Determine the Variance \(\operatorname{Var}(X)\)

The variance of a random variable \(X\), denoted by \(\operatorname{Var}(X)\), measures the spread of its values. The formula is \(\operatorname{Var}(X) = E((X - E(X))^2)\). Since \(X = 3\) and \(E(X) = 3\), we have \(X - E(X) = 0\) for all \(\omega\). Therefore, \((X - E(X))^2 = 0\), which gives \(\operatorname{Var}(X) = E(0) = 0\).

Key concepts

Expectation

Expectation, often denoted as \( E(X) \), is a fundamental concept in probability and statistics that represents the average or mean value of a random variable. It gives us a way to summarize the central tendency of the random variable's possible outcomes.
The idea is similar to calculating the average score of students in a class, where each score is multiplied by the probability of it occurring, then summed up. If you have a random variable \( X \) defined on a sample space and it constantly returns a fixed value like in our exercise, its expectation is simply that fixed value multiplied by probability, which in our case is 1.
This results in

• For a constant random variable: \( E(X) = X \times 1 = X \)
• If \( X = 3 \) consistently: \( E(X) = 3 \)

Thus, the expectation is a "predictable" value that gives a sense of the random variable's overall tendency.

Variance

Variance, represented as \( \operatorname{Var}(X) \), quantifies the variability or spread in the possible values of a random variable around its mean \( E(X) \). It's like measuring how the students' scores differ from their average score and then averaging that difference.
Mathematically, variance is calculated by taking the average of the squared differences from the mean. This is why the formula \( \operatorname{Var}(X) = E((X - E(X))^2) \) is so important. It captures fluctuations by focusing more on larger deviations.
When a random variable gives a constant outcome, like \( X = 3 \) in our exercise, every outcome is equals the mean \( E(X) \), leading to

• The difference \( X - E(X) = 0 \)
• Squaring this difference gives \( (X - E(X))^2 = 0 \)

As a result, the variance becomes zero, illustrating no variability.

Sample Space

The concept of a sample space, often denoted as \( \Omega \), is an essential aspect of probability theory. It includes all the possible outcomes of a random experiment.
Think of it as the entire collection of potential occurrences for any random setting. Whether you're flipping a coin or rolling a die, the sample space lists every eventuality.
In our exercise, the sample space contains all outcomes

• Each outcome is assigned the same value by the random variable
• Specifically in the exercise, every \( \omega \) in the sample space \( \Omega \) is mapped to 3

Understanding the sample space is crucial because it sets the stage for analyzing any random variable and its related properties, like expectation and variance. Knowing the structure of \( \Omega \) helps to comprehend how a random process unfolds and interacts with assigned values.