Math 464 Homework 7 Spring 2013 The Following Assignment Is

Math 464homework 7spring 2013the Following Assignment Is To Be Turned

The assignment involves analyzing various discrete random variables and their distributions through probability calculations, independence assessments, expected values, and variances. It includes problems related to dice experiments, joint probability mass functions, properties of Poisson and geometric distributions, and combined random variable analyses. Specifically, students are asked to find joint distributions, expected values such as E(XY), assess independence, determine the distribution of sums of independent variables, and compute variances, all within the context of specific experiments with dice, coins, and general distributions.

Paper For Above instruction

In this comprehensive analysis, we explore diverse problems traditionally encountered in the study of discrete random variables. These problems involve calculating joint probability distributions, examining independence, and calculating expectations and variances for complex random variables, often involving sums or functions of base distributions such as Poisson, geometric, and Bernoulli variables.

Starting with the problem involving dice rolls, the primary task is to model the distributions of the number of odd and even dice (X and Y), as well as the number of dice showing 3 or 4 (Z). When rolling two fair four-sided dice, each die independently takes values from 1 to 4 with equal probability. The variables X, Y, and Z can assume values in the set {0, 1, 2}, corresponding to the counts of particular outcomes among the two dice.

For part (a), the joint probability distribution of X and Y (fX,Y(x,y)) can be tabulated based on all possible outcomes of the two dice. The probability that a die is odd (values 1, 3) or even (values 2, 4), or showing a 3 or 4, informs the joint distribution, which can be calculated by enumerating each possible outcome and aggregating according to the values of X and Y. Given the symmetry and independence of the dice, the probabilities can be systematically tabulated, demonstrating the behavior of joint probabilities as well as their sums consistent with the total probability axioms.

Part (b) asks whether X and Y are independent. Since X counts odd outcomes and Y counts even outcomes, the two are dependent because an outcome cannot be both odd and even simultaneously. Formal checks involve verifying whether the joint pmf factorizes into marginal pmfs.

Part (c) involves calculating the expected value of the product XY. This expectation provides insights into joint variability and correlation, often requiring summing over all possible pairs weighted by their joint probabilities.

The second problem involves a simple joint distribution where the pmf is given by fX,Y(x,y) = α(x + y + 1) for x, y in {0, 1, 2}. To find E(XY) and E(Y), one uses the normalization condition to determine α. Once α is known, expectations are computed by summing over all possible (x, y) pairs, multiplying by the respective values and probabilities.

The third problem explores the sums and differences of independent variables. Given the moments of X and Y, the means and variances of linear combinations such as Z = 2X + Y and W = Y² - 2Y X² are calculated using properties of independence and moments. Variance calculations utilize the formulas Var(aX + bY) = a²Var(X) + b²Var(Y) for independent variables.

The fourth problem demonstrates the Poisson distribution's closure under addition. When X and Y are independent Poisson with parameters λ and μ respectively, their sum Z = X + Y also follows a Poisson distribution with parameter λ + μ. This property results from the convolution of Poisson pmfs, which can be proved via generating functions or combinatorial sums involving binomial coefficients.

The fifth problem involves a two-stage experiment with an unfair coin. The first stage involves flipping until a head appears; the second, until a tail appears. Random variables X (total flips) and Y (difference between numbers of heads and tails) are analyzed for their expectation and variance, with particular attention to their dependence on Bernoulli trials and geometric distributions. Decomposition into independent components simplifies the computations.

The sixth problem considers the joint distribution of two geometric random variables, X and Y, with specified expectations. Using the independence assumption and the known moments, the joint pmf of the minimum and maximum, W and Z, is derived, leveraging properties of order statistics in discrete distributions.

Problems seven and eight extend the themes of sums of independent variables, Poisson distributions, and Bernoulli trials, emphasizing the understanding of sample sums and the independence of counts when parameters are random. Specifically, the last problem involves mixed scenarios where the total counts depend on a Poisson number of Bernoulli trials, illustrating the Poisson compounding and independence properties in stochastic processes.

Throughout these exercises, fundamental concepts such as calculation of joint and marginal probabilities, expectation, variance, independence, and distributional properties are illustrated. The techniques employed include enumeration of outcomes, utilization of generating functions, properties of well-known distributions, and algebraic manipulation of moments. These problems collectively deepen understanding of the structure and behavior of discrete random variables in probabilistic modeling.

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