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Chapter 3: Problem 1

(a) Let \(X\) and \(Y\) be independent discrete random variables, and let \(g, h: \mathrm{R} \rightarrow \mathbb{R}\). Show that \(g(X)\) and \(h(Y)\) are independent. (b) Show that two discrete random variables \(X\) and \(Y\) are independent if and only if \(f_{X, Y}(x, y)=\) \(f X(x) f y(y)\) for all \(x, y \in \mathbb{R}\). (c) More generally, show that \(X\) and \(Y\) are independent if and only if \(f_{X, y}(x, y)\) can be factorized as the product \(g(x) h(y)\) of a function of \(x\) alone and a function of \(y\) alone. 2\. Show that if \(\operatorname{var}(X)=0\) then \(X\) is almost surely constant; that is, there exists \(a \in \mathbb{R}\) such that \(\mathbb{P}(X=a)=1\). (First show that if \(\mathrm{E}\left(X^{2}\right)=0\) then \(\left.\mathbb{P}(X=0)=1 .\right)\)

Short Answer

Expert verified
(a) Functions are independent as inputs are independent. (b) Use factorization of joint PMFs. (c) Factorization checks independence. (2) Zero variance implies constant variable.

Step by step solution

01

Establish Independence for g(X) and h(Y)

Since \(X\) and \(Y\) are independent random variables, their joint probability function \(f_{X,Y}(x,y)\) can be expressed as the product of their marginal densities: \(f_X(x)\) and \(f_Y(y)\). Therefore, probability functions for their respective transformations also factorize as independent because any function \(g(X)\) depends only on \(X\) and \(h(Y)\) depends only on \(Y\). Thus, \(g(X)\) and \(h(Y)\) are independent by the definition of independence.
02

Prove Independence Criterion for Discrete Random Variables

By definition, two discrete random variables \(X\) and \(Y\) are independent if the joint probability mass function \(f_{X,Y}(x,y)\) equals the product of the marginal distributions \(f_X(x)\) and \(f_Y(y)\) for all \(x, y\). This relationship confirms that the occurrence of \(X\) has no effect on the occurrence of \(Y\), and vice versa.
03

General Factorization Proof for Independence

For independence, the joint probability distribution \(f_{X,Y}(x,y)\) must factorize into a product of a function of \(x\) only and a function of \(y\) only, i.e., \(f_{X,Y}(x,y) = g(x)h(y)\). This expresses that \(X\) and \(Y\) do not influence each other's probability structures, thus exhibiting statistical independence if such factorization exists.
04

Zero Variance Implies Constant Variable

The variance of \(X\) is defined as \(\operatorname{var}(X) = E[(X - E[X])^2]\). If \(\operatorname{var}(X) = 0\), then \(E[(X - E[X])^2] = 0\), implying that \(X = E[X]\) almost surely, hence \(X = a\) for some constant \(a\) with probability 1.
05

Expectation of Squared Zero Implies Certainty

If \(E(X^2) = 0\), then \(P(X^2 > 0) = 0\), which means \(X\) must be zero almost surely, so \(P(X = 0) = 1\), meaning that \(X\) must be constant at 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Random Variables
Discrete random variables play a crucial role in probability and statistics. They are variables that can take a countable number of distinct values. Unlike continuous random variables, which can take any value within a given range, discrete random variables have specific, separate values. Common examples include the number of heads in a series of coin tosses, the roll of a die, or the number of students in a class.

When working with discrete random variables, probabilities are assigned to each possible value that the variable can take. This assignment is represented by a probability mass function (PMF). The PMF quantitatively describes the likelihood of each outcome. If we have a discrete random variable \(X\), the probability that \(X\) takes a value \(x\) is denoted by \(P(X = x)\). Importantly, the sum of all probabilities within the PMF must equal 1.
Joint Probability Distribution
In probability theory, a joint probability distribution is used when dealing with two or more random variables simultaneously. Specifically, in the context of two discrete random variables \(X\) and \(Y\), their joint probability distribution is characterized by the joint probability mass function \(f_{X,Y}(x, y)\). This function provides the probability that \(X\) is equal to \(x\) and \(Y\) is equal to \(y\) simultaneously.

Joint probability distributions can be understood as an extension of single-variable distributions to multiple dimensions. If the variables are independent, the joint distribution becomes the product of their marginal distributions: \(f_{X,Y}(x, y) = f_X(x) \cdot f_Y(y)\). This independence indicates that the occurrence of one variable does not affect the occurrence of the other, a vital concept in multivariate probability.
Variance and Expectation
Variance and expectation are fundamental concepts in statistics that help describe the distribution of a random variable. The expectation or expected value of a random variable \(X\), denoted as \(E[X]\), gives the long-term average or mean value of the variable upon numerous trials or samples. It is calculated by summing all possible values of \(X\) weighted by their probabilities.

Variance, denoted by \(\operatorname{var}(X)\), measures how much the values of \(X\) deviate or display variability around the expected value. It is calculated as \(E[(X - E[X])^2]\). A key property is that if \(\operatorname{var}(X) = 0\), the variable \(X\) is constant almost surely, meaning there is no variation, and \(X = a\) for some constant \(a\) with probability 1.
Factorization in Probability
Factorization in probability is an essential concept used to determine the independence of random variables. It involves expressing the joint probability distribution of two variables \(X\) and \(Y\) as the product of two separate functions, one only dependent on \(X\) and the other only on \(Y\). This expression can take the form \(f_{X,Y}(x, y) = g(x) \cdot h(y)\).

The ability to factorize the joint probability distribution in such a manner indicates that the random variables are independent. This is because it shows that the behavior or occurrence of one variable does not influence the other. Hence, such factorization is a powerful tool for identifying and proving independence in probability theory.

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Most popular questions from this chapter

A coin is tossed repeatedly, heads turning up with probability \(p\) on each toss. Player \(\mathrm{A}\) wins the game if \(m\) heads appear before \(n\) tails have appeared, and player B wins otherwise. Let \(p_{m n}\) be the probability that \(A\) wins the game. Set up a difference equation for the \(p_{m n}\). What are the boundary conditions?

Let \(T\) be the time which elapses before a simple random walk is absorbed at either of the absorbing barriers at 0 and \(N\), having started at \(k\) where \(0 \leq k \leq N\). Show that \(P(T<\infty)=1\) and \(\mathbb{E}\left(T^{k}\right)<\infty\) for all \(k \geq 1\).

In a certain style of detective fiction, the sleuth is required to declare "the criminal has the unusual characteristics ... ; find this person and you have your man". Assume that any given individual has these unusual charncteristics with probability \(10^{-7}\) independently of all other individuals, and that the city in question contains \(10^{7}\) inhabitants. Calculate the expected number of such people in the city. (a) Given that the police inspector finds such a person, what is the probability that there is at least one other? (b) If the inspector finds two such people, what is the probability that there is at least one more? (c) How many such people need be found before the inspector can be reasonably confident that he has found them all? (d) For the given population, how improbable should the characteristics of the criminal be, in order that he (or she) be specified uniquely?

Let \(\mathbf{X}=\left(X_{1}, X_{2}, \ldots, X_{n}\right)\) be a vector of random variables. The covariance matrix \(\mathbf{V}(\mathbf{X})\) of \(\mathbf{X}\) is defined to be the symmetric \(n\) by \(n\) matrix with entries \(\left(v_{i j}: 1 \leq i, j \leq n\right)\) given by \(v_{i j}=\operatorname{cov}\left(X_{i}, X_{j}\right)\). Show that \(|\mathbf{V}(\mathbf{X})|=0\) if and only if the \(X_{t}\) are linearly dependent with probability one, in that \(\mathbb{P}\left(a_{1} X_{1}+a_{2} X_{2}+\cdots+a_{n} X_{n}=b\right)=1\) for some \(\mathbf{a}\) and \(\dot{b} .(\mid \mathbf{V}]\) denotes the determinant of \(\mathbf{V} .\) )

Of the \(2 n\) people in a given collection of \(n\) couples, exactly \(m\) die. Assuming that the \(m\) have been picked at random, find the mean number of surviving couples. This problem was formulated by Daniel Bemoulli in 1768 .
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