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Showing posts from September, 2014

### Probability Theory Problems

Let's have fun on probability theory, here is my first problem set in the said subject.

Problems It was noted that statisticians who follow the deFinetti school do not accept the Axiom of Countable Additivity, instead adhering to the Axiom of Finite Additivity. Show that the Axiom of Countable Additivity implies Finite Additivity.Although, by itself, the Axiom of Finite Additivity does not imply Countable Additivity, suppose we supplement it with the following. Let $A_1\supset A_2\supset\cdots\supset A_n\supset \cdots$ be an infinite sequence of nested sets whose limit is the empty set, which we denote by $A_n\downarrow\emptyset$. Consider the following:

Axiom of Continuity: If $A_n\downarrow\emptyset$, then $P(A_n)\rightarrow 0$

### Translation Invariant of Lebesgue Outer Measure

Another proving problem, this time on Real Analysis.

Problem Prove that the Lebesgue outer measure is translation invariant. (Use the property that, the length of an interval $l$ is translation invariant.)
SolutionProof. The outer measure is translation invariant if for $y\in \mathbb{R}$, $$\nonumber \mu^{*}(A)=\mu^{*}(A+y)$$ Hence, we need to show that Case 1: $\mu^{*}(A)\leq \mu^{*}(A+y)$; and Case 2: $\mu^{*}(A+y)\leq \mu^{*}(A)$.

Case 1: Consider a countable collection $\{I_n\}_{n=1}^{\infty}$, and let $$\nonumber W = \left\{\displaystyle\sum_{n=1}^{\infty}l(I_n)\mid A\subseteq\displaystyle\bigcup_{n=1}^{\infty}I_n\right\}$$ Then the outer measure of $A$ is, $$\nonumber \mu^{*}(A)=\inf\,\{W\}.$$

### R: Image Analysis using EBImage

Currently, I am taking Statistics for Image Analysis on my masteral, and have been exploring this topic in R. One package that has the capability in this field is the EBImage from Bioconductor, which will be showcased in this post.

Installation
For those using Ubuntu, you may likely to encounter this error:

It has something to do with the tiff.hC header file, but it's not that serious since mytechscribblings has an effective solution for this, do check that out.

Importing Data To import a raw image, consider the following codes:

### Monotonic Sequential Continuity

This problem is the continuation of my previous post on Monotonic Sequence.

Problem Prove the following: If $A_k$ is monotone, then $$\mathrm{P}\left(\displaystyle\lim_{n\to\infty} A_n\right)=\displaystyle\lim_{n\to \infty}\mathrm{P}(A_n).$$
SolutionProof. If $\{A_k\}$ is monotone, then $$\nonumber \mathrm{P}\left(\lim_{n\to \infty} A_n\right) = \begin{cases} \displaystyle\mathrm{P}\left(\bigcup_{k=1}^\infty A_k\right)&\text{if}\;\{A_k\}\;\text{is expanding}\\ \displaystyle\mathrm{P}\left(\bigcap_{k=1}^\infty A_k\right)&\text{if}\;\{A_k\}\;\text{is contracting} \end{cases}.$$ So if $A_k$ is expanding, then we can write $\displaystyle\bigcup_{k=1}^\infty A_k$ as disjoint unions, \begin{eqnarray} \displaystyle\bigcup_{k=1}^\infty A_k &=& A_1\cup (A_2\cap A_1^c)\cup (A_3\cap A_2^c)\cup \cdots\nonumber\\ &=& A_1\cup (A_2\backslash A_1)\cup (A_3\backslash A_2)\cup \cdots\nonumber \end{eqnarray}