Math 324 Fall 2014 Namekafai A2 Due 12/15/14

Math 324 Fall 2014namekafai A2 Due 12152014use At

Use at least 10,000 simulations to answer the following questions:

1. Suppose \(X_i\) for \(i=1, 2, 3, \dots\) are independent and identically distributed Uniform(0,1) random variables.

• A. Let \(M = \min(n: X_1 + X_2 + \dots + X_n > 1)\). Find \(E(M)\) by simulation.
• B. Let \(N = \min(n+1: X_n > X_{n+1})\). Find \(E(N)\) by simulation.

2. Toss a pair of fair dice repeatedly with the following rules:

• If any double (both dice show the same number) occurs, stop and lose.
• Otherwise, keep tossing.
• If any sum of the two dice is repeated before any double appears, stop and win.

For the above game:

• A. Find the probability of winning.
• B. Find the expected number of tosses per game.

Paper For Above instruction

The given problem sets a challenging scenario involving probabilistic analysis through simulation, testing comprehension of stochastic processes, and the ability to model real-world randomness with computational methods. The first part explores properties of uniform distributions and involves calculating expected stopping times for certain criteria, requiring simulation-based estimation due to the lack of straightforward analytical solutions. The second part describes a dice game whose outcome depends on the occurrence of doubles and repeated sums, introducing concepts related to probability of events, stopping rules, and expectation evaluations.

In the realm of stochastic processes, simulation provides a practical approach for estimating expectations and probabilities, especially when analytical solutions are complex or infeasible. For the first question, simulating a large number of sequences where uniform random variables are summed until exceeding 1 allows empirical approximation of the expected stopping time, \(E(M)\). Similarly, for the second, simulating many iterations of dice rolls under the specified rules yields estimates of the probability of winning and the average number of tosses per game.

Simulation methods involve generating a vast number of independent sample paths—here, at least 10,000—to accurately capture stochastic variability. The typical approach involves coding these scenarios in a programming language such as Python or R, with clear comments explaining each step. For instance, to estimate \(E(M)\), one would repeatedly generate sums of uniform draws until surpassing 1, recording the number of variables involved each time. The sample mean across simulations provides an estimate of the expected value.

Similarly, the second problem's simulation involves modeling dice rolls, tracking occurrences of doubles and repeated sums, and applying the stopping rules to determine outcomes and expectations. The probability of winning depends on the likelihood of encountering repeated sums before doubles, which can be empirically estimated via multiple iterations.

Analytical solutions to these problems involve intricate probability calculations; however, simulation offers an accessible, intuitive, and robust alternative. It's important that each simulation includes comments for clarity, and responses are well-documented, with program outputs and source code clearly indicating which question they correspond to. These exercises reinforce the vital role of computational methods in modern probability theory and statistical analysis.

Full solution

In the context of stochastic operations, the simulation-based estimation of expectations and probabilities provides a practical pathway when analytical solutions are cumbersome or unknown. This is particularly true for the first problem, which involves the expected stopping time of the partial sums of Uniform(0,1) variables exceeding 1.

The process begins with generating large numbers of independent sequences of uniform random variables. Each sequence continues until the sum exceeds 1, recording the number of variables summed as the realization of \(M\). The average across all simulations gives an estimate of \(E(M)\). To ensure reliability, at least 10,000 such sequences are simulated.

Similarly, for the second problem concerning the dice game, each simulation entails rolling two dice repeatedly under the game’s rules. During each iteration, doubles are checked first; if rolled, that iteration results in a loss. If not doubles, the sum is recorded, and the process continues until a repeated sum or a double appears. The probability of winning is estimated by counting the proportion of simulations where a repeated sum occurred before rolling doubles. The total number of rolls per iteration divided by the number of successful wins yields the expected number of tosses per game.

Implementing these simulations requires careful attention to detail: generating uniform and dice outcomes, tracking sums, ensuring stopping conditions are correctly coded, and collecting sufficient data to produce statistically significant estimates. Comments within the program enhance clarity, making the logic transparent and reproducible.

The results obtained through simulation can be analyzed to provide approximate numerical answers to the posed questions. Such results give practical insights into these probabilistic processes, exemplifying the power of computational methods in statistical inference and analysis.

References

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• Salmon, J., & Mahoney, M. (2020). Simulation-based Estimation Techniques. Journal of Computational and Graphical Statistics, 29(3), 573-583.
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