## Q

(These are some thoughts I got while reading the inspiring book by James D.Watson “Avoid boring people and other lessons from a life in science.” Maybe I will give a full summary on the lessons from the book later.)

## Mathematics and the Unexpected

(Just some thoughts while I read Mathematics and the Unexpected and Innumeracy: Mathematical Illiteracy and its consequence. Maybe some more detailed post later.)

## Probability and Computing: Chapter 7 Exercises

Exercise 7.12: Let $X_{n}$ be the sum of $n$ independent rolls of a fair dice. Show that, for any $k > 2$, $\lim_{n \rightarrow \infty}(X_{n} \text{is divisible by k}) = \frac{1}{k}$.

## An exercise about Markov Chains

Lately  I have stopped reading “Probability and computing”, since I found some gaps in the exposition of the text, especially at chapter 7 “Markov Chains and Random Walks”-the authors left undefined some terminologies such as absorption. Certainly this is not the text for anybody who has little background of probability and want to learn it rigorously (though it is a good introductory text for randomized algorithm). So I bought “Markov Chains” of James Norris.
Exercises in “Markov Chains” are easy (at least for the first chapter ), though there are some problems that I am not quite sure. Here is one of them:

## Linear algebra viewpoint of function of discrete random variable

Suppose $X$ is a discrete random variable can take on $n$ values $x_{1} < x_{2} <\ldots with probability $p_{x_{1}},p_{x_{2}},\ldots ,p_{x_{n}}$; and $Y$ is a random variable defined by $Y = g(X)$.

## Probability and Computing: Chapter 3 Exercises (cont.3)

Exercise 3.21: A fixed point of a permutation $\pi : [1,n] \to [1,n]$ is a value for which $\pi (x) = x$. Find the variance in the number of fixed points of a permutation chosen uniformly at random from all permutations. (Hint: Let $X_{i}$ be $1$ if $\pi (i) =i$, so that $\sum_{i=1}^{n}X_{i}$ is the number of fixed points. You cannot use linearity to find $Var[\sum_{i=1}^{n}X_{i}]$, but you can calculate it directly.)

## Probability and Computing: Chapter 3 Exercises (cont.2)

Exercise 3.15: Let the random variable $X$ be representable as a sum of random variables $X=\sum_{i=1}^{n}X_{i}$. Show that, if $E[X_{i}X_{j}]=E[X_{i}]E[X_{j}]$ for every pair of $i$ and $j$ with $1 \leq i , then $Var[X] = \sum_{i=1}^{n}Var[X_{i}]$.