Compute distribution of dice maximum

Q: Compute distribution of dice maximum

The question evaluates understanding of discrete probability and order statistics, specifically deriving PMF and CDF for maxima, computing moments (expectation and variance), and generalizing results for independent and biased six-sided dice.

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Question

Max of Six-Sided Dice — Distribution and Moments

You roll fair six-sided dice and consider the maximum outcome.

Assume outcomes are in {1, 2, 3, 4, 5, 6}. For parts (a)–(b), you roll two independent fair dice. For part (c), you roll n i.i.d. fair dice. For part (d), you roll n i.i.d. biased dice where face i has probability p_i (with $\sum_{i=1}^6 p_i = 1$ ).

(a) Derive the PMF and CDF of the maximum for two fair dice.

(b) Compute the expected value and variance for the maximum of two fair dice.

(c) Generalize the PMF and the expected value to n i.i.d. fair six-sided dice.

(d) For n i.i.d. biased dice with face probabilities $\{p_i\}_{i=1}^6$ , express $\mathbb{P}(\max \le k)$ and $\mathbb{E}[\max]$ in terms of $\{p_i\}$ .

Compute distribution of dice maximum

Max of Six-Sided Dice — Distribution and Moments

Solution

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Compute distribution of dice maximum

Overview

Max of Six-Sided Dice — Distribution and Moments

Solution

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