Articles of probability

Combinatorics/Probability – Why does this equation work?

Five distinct numbers are randomly distributed to players numbered $1$ through $5$. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players $1$ and $2$ compare their numbers; the winner then compares her number with that of player $3$, and so on. Let $X$ denote the number […]

Are X and |X| independent? where $f(x)=\exp(-|x|)/2$

The density function of $X$ is $f(x)=\frac{e^{-|x|}}2, x\in(-\infty, +\infty)$. Are $X$ and $|X|$ independent? My thought is: Let $Y=|X|$, so $f(y)=e^{-y},y\in(0,+\infty)$ Then try $P(X\le x,Y\le y)$, I cannot go further, can anyone help? Edit:Thank you guys and please allow me to extend my question, is there a general way to determine the independence of 2 […]

Help determining best strategy for game?

Imagine you have 5 different territories on a game board. Suppose two players each have 100 soldiers. They both independently distribute their 100 soldiers throughout the 5 territories. The two players then reveal where they placed their soldiers; Whoever has the most soldiers on each territory takes control of that territory. Which-ever player has the […]

Are there probability density functions with the following properties?

Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions: $$\mathscr{G}_0=\left\{g_0:\int_{\mathbb{R}}\log\left(\frac{g_0(y)}{f_0(y)}\right)g_0(y)\mathrm{d}y\leq \epsilon_0\right\} $$ and $$\mathscr{G}_1=\left\{g_1:\int_{\mathbb{R}}\log\left(\frac{g_1(y)}{f_1(y)}\right)g_1(y)\mathrm{d}y\leq \epsilon_1\right\} $$ for some sufficiently small $\epsilon_0$ and $\epsilon_1$ such that $\mathscr{G}_0$ and $\mathscr{G}_1$ are also distinct. In other words any density $g_0\in \mathscr{G}_0$ is not an element of […]

How to show that these random variables are pairwise independent?

Given the arrays $C=[C_1,C_2,…,C_N]$ and $S=[S_1,S_2,…,S_N]$ of lengths $N$ with elements that are discrete iid uniform distributed with equal probability (p=1/2) of being $\pm$ 1 Consider the random variables (for a given $l, n, m$): $W=C_lC_mC_n$ $X=S_lS_mC_n$ $Y=C_lS_mS_n$ $Z=S_lC_mS_n$ It can be shown that these random variables ($W, X, Y, Z$) are zero mean, uniform […]

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I’m trying to find the distribution of the “range”: $$R_{t} = \sup_{0 \leq s \leq t} B_s – \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The reflection principle gives, for $a > 0$, $P(\overline{M_t} \geq a) = 2 P(B_t \geq a)$ (as stated by @A.S. as comment on […]

Simplification / Concentration Results / Bounds for the Expectation of the Minimum of Two Correlated Random Variables

Question, Please provide any simplifications / bounds / concentration results for the expectation of the minimum of two correlated random variables. $$E\left[\min\left(X,Y\right)\right]=\text{??}$$ $$E\left[\min\left(X,Y\right)\right]\leq\text{??}$$ $$E\left[\min\left(X,Y\right)\right]\geq\text{??}$$ Assumptions, We can assume that $X$, $Y$ have a covariance $\sigma_{XY}$ and are from general distribution, $G\left(x\right)$ and $F\left(y\right)$ respectively. We can assume that the mean, variance and higher moments exist […]

Why doesn't this conditional probability equation hold

This is probably a very stupid question but I can’t wrap my head around it. $$ P(B \cap A) = P(A \cap B) = P(B \mid A)\cdot P(A) + P(A \mid B)\cdot P(B) $$ Can someone explain intuitively why the above isn’t true? What I am essentially saying here is that A and B both […]

Convolution Theorem and Marginal Density Intuition.

In terms of marginal density, how does one know that summing over the $x$ (or rather along the linear line) values for the joint density of $(x,z-x)$ give us the density function of $z$? More importantly, can someone explain this intuitively? (aside from proofs)

Which model to use ? (probability problem)

The following was the problem that I was working on. As a part of the underwriting process for insurance, each prospective policyholder is tested for high blood pressure. Let X represent the number of tests completed when the first person with high blood pressure is found. The expected value of X is 12.5 Calculate the […]