Math 464 Homework 4 Spring 2013 The Following Assignment Is

Math 464homework 4spring 2013the Following Assignment Is To Be Turned

This assignment involves analyzing properties of discrete random variables, proving certain probability sum properties, calculating probabilities for specific distributions such as Poisson and binomial, and computing expectations for these distributions. The tasks include demonstrating that transformations of discrete random variables remain discrete, verifying probability distributions sum to one, deriving probability formulas, and finding expected values both through direct calculation and summation techniques. Additionally, the assignment requires studying the mode of binomial distributions, exploring functions of random variables, and checking the validity of probability mass functions for specific cases.

Paper For Above instruction

Discrete random variables are foundational in probability theory, and understanding their properties is crucial for advanced statistical analysis. The assignment begins with demonstrating that the transformation of a discrete random variable via a function remains a discrete random variable. This involves leveraging the fact that the image of a countable set under a measurable function stays countable, and the probability measure assigns probabilities accordingly.

Specifically, let \(X:\Omega \to \mathbb{R}\) be a discrete random variable, meaning its range is countable, say \(\{x_i\}\). Define a new random variable \(Y = g(X)\), where \(g : \mathbb{R} \to \mathbb{R}\). To prove \(Y\) is discrete, we observe that \(Y\) takes values in the set \(g(\{x_i\})\), which is at most countable since \(g\) maps a countable set to a countable set (or possibly finite). The probability that \(Y\) takes a value \(y\) in its range is the sum of the probabilities \(P(X = x_i)\) over all \(x_i\) such that \(g(x_i) = y\). Since these sums are countable and the sum of probabilities over the entire range of \(X\) is 1, \(Y\) is also a discrete random variable.

Next, the geometric distribution's probability mass function (pmf), \(P(X=k) = p(1-p)^{k-1}\) for \(k \geq 1\), is verified by summing over all possible \(k\). Summing from \(k=1\) to infinity, the geometric series converges to 1, confirming that this pmf is valid. The geometric distribution models the number of trials up to and including the first success in independent Bernoulli trials with success probability \(p\).

The properties of the Poisson distribution are explored next. For a Poisson random variable \(X \sim Poisson(\lambda)\), the probabilities \(P(2 \leq X \leq 4)\), \(P(X \geq 5)\), and \(P(X \text{ is even})\) are computed exactly using the pmf \(P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}\). For \(\lambda=2\), numerical approximations are provided with three decimal places accuracy. These computations are vital for understanding the distribution's shape and tail behavior.

Furthermore, the expectation of a discrete random variable \(X\) with support \(\{0,1,2,\dots\}\) is expressed as an infinite sum of tail probabilities: \(E(X) = \sum_{k=0}^\infty P(X > k)\). This representation connects the expectation directly to the tail probabilities and provides an alternative to the sum over \(k \cdot P(X=k)\). Using this, the expected value of a geometric random variable with parameter \(p\) is derived as \(1/p\), aligning with standard results.

The ratio of probabilities \(P(X=k-1)/P(X=k)\) for a binomial distribution \(X \sim Bin(n,p)\) is computed to analyze the mode of the distribution. It is shown that this ratio is less than 1 precisely when \(k

Investigation into functions of \(X\), particularly \(g(x) = x(x-1)\), is conducted for a Poisson distributed \(X\). Computing \(E(g(X))\) involves summing over the pmf weighted by \(x(x-1)\), which simplifies by recognizing the factorial moments of the Poisson distribution. This illustrates how moments of the distribution can be derived from generating functions or summation techniques.

Finally, the question assesses a proposed probability mass function \(P(X=n) = 1/[n(n+1)]\) for \(n \geq 1\), and examines whether it defines a valid distribution. Summing over all \(n\), it is confirmed that the total sums to 1, making \(X\) a valid discrete random variable. Its expected value is then computed using the sum \(\sum_{n=1}^\infty n \cdot P(X=n)\), leading to a specific numerical result.

References

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