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please show the detailed steps and proof for all questions 1. Let $f:\{(x, y): 0...

Oct 12, 2024

please show the detailed steps and proof for all questions

Solution by Steps

For Problem 1:

step 1

We apply the Weierstrass approximation theorem, which states that any continuous function on a closed interval can be uniformly approximated by polynomials

step 2

Given

f: \{(x,y): 0 \leq x, y \leq 1\} \to \mathbb{R}

is continuous, for every \varepsilon > 0 , there exists a polynomial

P(x,y)

such that: \max_{0<x,y<1} |f(x,y) - P(x,y)| < \varepsilon

step 3

We can construct

P(x,y)

as a sum of bivariate monomials:

P(x,y) = \sum_{i,j} a_{i,j} x^i y^j

where the coefficients

a_{i,j}

are chosen to minimize the maximum error

Answer

A bivariate polynomial

P(x,y)

can be constructed to approximate

f(x,y)

within any desired

\varepsilon

For Problem 2:

step 1

We start with the law of large numbers, which states that for i.i.d. random variables

X_1, X_2, \ldots

, the sample mean converges in probability to the expected value

step 2

Since

E(X) = \infty

, we know that the average

\frac{S_n}{n}

cannot converge to any finite M > 0

step 3

Therefore, we conclude that:

P\left(\frac{S_n}{n} \leq M\right) \to 0 \text{ as } n \to \infty

Answer

The probability

P\left(\frac{S_n}{n} \leq M\right)

approaches 0 as

n

increases.

For Problem 3:

step 1

We start with the given density function

f_X(x) = \frac{2}{\pi(1+x^2)}

, which is the density of a Cauchy distribution

step 2

We need to find

b_n

such that

\frac{S_n}{b_n} \xrightarrow{p} 1

. By the weak law of large numbers, we can set

b_n = n

since

E(X)

is undefined

step 3

We verify that: \frac{E(X; X < b_n)}{b_n} \to 1 holds true, confirming that

b_n

can be chosen as

n

Answer

The sequence

b_n

can be chosen as

n

for convergence in probability.

Key Concept

The Weierstrass approximation theorem, law of large numbers, and properties of Cauchy distribution are key concepts in these problems.

Explanation

The solutions demonstrate how to approximate functions with polynomials, analyze convergence of sample means, and utilize properties of specific distributions.