Statistical Inference I J Lee Assignment 1 Problem 1 Exam

Statistical Inference I J Lee Assignment 1problem 1 An Exam Has 10

This assignment involves multiple statistical problems covering probability, random variables, and inference. The problems require calculating probabilities, distributions, expectations, and understanding concepts related to events, conditioning, and probability distributions. Specifically, the tasks include analyzing binomial probabilities, composite events, joint probabilities, conditional probabilities, and properties of random variables with specified distributions, as well as applying statistical reasoning to real-world scenarios such as traffic, books, dice, and experiments.

Paper For Above instruction

The assignment encompasses ten distinct problems, each focusing on fundamental concepts in probability and statistics. The first problem examines binomial distributions involving guessing questions on an exam with and without prior knowledge, requiring calculation of probabilities for at least five correct answers. This involves understanding Bernoulli trials, binomial probability mass functions (pmf), and cumulative probabilities.

In the second problem, the focus shifts to mutually exclusive events with given probabilities. It involves calculating the probability of union, difference, and intersection of these events, applying basic probability rules such as the additive and multiplicative principles.

The third problem explores the joint probability bounds based on marginal probabilities of two events. It requests the calculation of maximum and minimum possible overlaps (intersections), and the bounds for the union, relying on properties of probability measures and set theory.

The fourth problem involves sampling cars from a row at traffic lights, modeled as a random arrangement. It specifically asks for the probability that a particular position (fifth in line) is occupied by a Mercedes, requiring an understanding of uniform sample spaces and combinatorial probabilities.

Similarly, the fifth problem deals with receiving boxes of books from various publishers over a month. The goal is to compute the probability that the last two received boxes originate from the same publisher, assuming equal likelihood. This problem involves defining the sample space for the last two boxes, considering all possible publisher pairs, and calculating favorable outcomes for the same publisher.

Problem six examines conditional probability involving dice rolls, focusing on the likelihood that at least one die shows a 6, given that the dice land on different numbers. This necessitates understanding joint distributions and conditional probability calculations.

In problem seven, the scenario involves demographic data about color-blindness in males and females. Given probabilities and equal population sizes, the task is to find the probability that a randomly selected color-blind person is male, involving Bayes’ theorem and conditional probability.

Further, problem eight pertains to randomly choosing integers under specified restrictions, requiring calculations of probabilities for specific pairings, such as the first number being 3 and the second exceeding 4, or both being less than 3, and both greater than 3—applying basic probability rules for distributed choices.

Problem nine presents a gamblers’ scenario with multiple coins—two fair coins and a two-headed coin—and asks for the probability of the coin being fair after various sequences of heads and tails. This problem involves Bayesian updating based on observed outcomes.

Finally, problem ten discusses properties of probabilities of events A and B, involving independence, mutual exclusivity, and conditional probabilities, asking for assessments of specific probability relationships based on provided data, testing understanding of foundational probability concepts.

Each problem collectively explores key statistical concepts including binomial distributions, joint and marginal probabilities, conditional probabilities, properties of random variables, and applying probabilistic reasoning to realistic scenarios, forming comprehensive practice in statistical inference and probability theory.

References

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• Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications (Vol. 1). Wiley.
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• Casella, G., & Berger, R. L. (2021). Statistical Inference (4th ed.). Cengage Learning.