Understanding the Expectation of a Random Variable with One Outcome

When dealing with the expectation of a random variable, it is crucial to understand how to compute and interpret it in different scenarios. In this article, we will explore a specific case where a random variable Z has only one outcome. We will walk through the calculation step-by-step and discuss the implications.

Case Study: A Random Variable with Only One Outcome

Consider the random variable Z which has only one outcome: 9. This means that Z takes the value 9 with a probability of 1. In mathematical terms:

P(Z 9) 1

The set of all outcomes, S, is given by:

S {9}

Calculating the Expectation

The expectation or expected value of a random variable, denoted as E[Z], is a measure of its central tendency or average value over a large number of repetitions of the experiment. In the case of a discrete random variable like Z, the expected value is calculated as the sum of the possible values multiplied by their respective probabilities.

For our random variable Z, the expected value can be computed using the formula:

E[Z] ∑s in S sP(Z s)

Substituting the given values into the formula:

E[Z] 9 * P(Z 9)

Since P(Z 9) 1, the equation simplifies to:

E[Z] 9 * 1

Therefore, the expected value is:

E[Z] 9

Interpreting the Expectation

Given that the random variable Z always takes the value 9, it is clear that the expected value is 9 without needing to compute anything. This is because the random variable is a constant with no variability. Hence, the expectation of a constant is the constant itself.

Thus, we can conclude that:

E[Z] 9

Other Considerations

While Z 9 is a straightforward case, this scenario can be useful in more complex models. For instance, it might represent a specific outcome in a political election model, where all potential candidates are eliminated except for one.

In general, understanding how to compute and interpret the expectation for different types of random variables is crucial in various fields such as finance, statistics, and data science.

Conclusion

The expectation of a random variable with one outcome is simply the value of that outcome. This concept is fundamental in probability theory and can be applied in many practical scenarios, including real-world data analysis and modeling.