(a) Suppose that we roll twice a fair k-sided die with the...

(a) Suppose that we roll twice a fair $k$-sided die with the numbers 1 through $k$ on the die's faces, obtaining values $X_{1}$ and $X_{2}$. What is $\mathbf{E}\left[\max \left(X_{1}, X_{2}\right)\right]$ ? What is $\mathbf{E}\left[\min \left(X_{1}, X_{2}\right)\right]$ ? (b) Show from your calculations in part (a) that $$ \mathbf{E}\left[\max \left(X_{1}, X_{2}\right)\right]+\mathbf{E}\left[\min \left(X_{1}, X_{2}\right)\right]=\mathbf{E}\left[X_{1}\right]+\mathbf{E}\left[X_{2}\right] . $$ (c) Explain why Eqn. (2.1) must be true by using the linearity of expectations instead of a direct computation.

(a) Suppose that we roll twice a fair $k$-sided die with the numbers 1 through $k$ on the die's faces, obtaining values $X_{1}$ and $X_{2}$. What is $\mathbf{E}\left[\max \left(X_{1}, X_{2}\right)\right]$ ? What is $\mathbf{E}\left[\min \left(X_{1}, X_{2}\right)\right]$ ?
(b) Show from your calculations in part (a) that
$$
\mathbf{E}\left[\max \left(X_{1}, X_{2}\right)\right]+\mathbf{E}\left[\min \left(X_{1}, X_{2}\right)\right]=\mathbf{E}\left[X_{1}\right]+\mathbf{E}\left[X_{2}\right] .
$$
(c) Explain why Eqn. (2.1) must be true by using the linearity of expectations instead of a direct computation.

Key Concepts

Uniform Distribution

The uniform distribution is a key concept in probability where every outcome in a finite set is equally likely. In the context of a die roll, each face of the die has the same probability of appearing, which simplifies the calculation of expected values by ensuring a consistent weighting for each outcome.

Expected Value

The expected value, or mathematical expectation, is a measure of the central tendency of a random variable, obtained by summing the products of each possible outcome with its corresponding probability. This concept is used to determine the average result of an experiment, such as rolling a die, over a large number of trials.

Linearity of Expectation

Linearity of expectation is a fundamental property that states the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of any dependence between the variables. This principle is especially useful in simplifying the computation of complex statistics, such as the sum of the max and min values in multiple trials.

Order Statistics

Order statistics involves the study of the properties of ordered random variables, like the maximum and minimum values from a set of independent random variables. In problems involving the maximum and minimum outcomes of die rolls, order statistics provides the framework to analyze and compute their expected values and understand their relationship to the overall expected outcomes.

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