Articles of probability

If I have that $X$ is a random variable satisfying $0\leq X \leq 1$, how can I show that $P\left(X \geq \frac{E(X)}{2}\right) \geq \frac{E(X)}{2}$?

If I have that $X$ is a random variable satisfying $0\leq X \leq 1$, how can I show that $P\left(X \geq \frac{E(X)}{2}\right) \geq \frac{E(X)}{2}$? I saw a footnote which gave a hint to split up $E(X)$ as two integrals over $\left[X<\frac{E(X)}{2}\right]$ and $\left[X\geq\frac{E(X)}{2}\right]$. However, I am not quite sure how to bound this integral. Would […]

Probability that a normal distribution is greater than two others

Given 3 independent variables with normal distributions, how can I calculate the probability that one of them will be greater than the other two simultaneously? So, how to calculate $P ((A>B) \bigcap (A>C))$ with $A=N(\mu_A,\sigma_A)$ $B=N(\mu_B,\sigma_B)$ $C=N(\mu_C,\sigma_C)$ I know how to calculate $P(A>B)$, since $P(A>B) = P(A-B>0)$, where $A-B$ has a normal distribution with $\mu […]

example of irreductible transient markov chain

Can anyone give me a simple example of an irreductible (all elements communicate) and transient markov chain? I can’t think of any such chain, yet it exists (but has to have an infinite number of elements) thanks

Uniformly distributed probability problem

May you have an idea for the following exercise I found from some olympiad. Each day you have to bring home one full can of water. To do so you go to the next well and make the can full. On the way home you loose a proportion which is uniformly distributed on [0,1]. Question: […]

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don’t have the knowledge on this matter to find out how they will evolve. I hope you help me here, with bibliography and teaching me some things. This is the system: I have a discrete distribution of probability of […]

Joint Probability involving Uniform RV

A man and a woman decide to meet at $12:30$. If the man arrives at a time uniformly distributed between $12:15 – 12:45$, and if the woman independently arrives at a time uniformly distributed between $12:00 – 1:00$, find the probability that the first to arrive waits no longer than $5$ minutes. Attempt: I can […]

PDF of the ratio of two independent Gamma random variables

Let $X \sim \operatorname{Gamma}(a,\lambda)$ and $Y \sim \operatorname{Gamma}(b,\lambda)$ being independent. Find the PDF of the ratio $W=X/Y$. I found $$ f_W(w) = \frac{\Gamma(a+b)}{\Gamma(a) + \Gamma(b)} \left(\frac{w}{w+1}\right)^a \left(\frac{1}{w+1}\right)^b \frac{1}{w} $$ So, $$ f_W(w) = \operatorname{dbeta}\left(\frac{w}{w+1}, a+1, b+1 \right) \frac{1}{w} $$ or $$ f_{X/Y}(x/y) = \operatorname{dbeta}\left(\frac{x}{x+y}, a+1, b+1 \right) \frac{y}{x} $$ Is there any story or interpretations […]

Traditional combination problem with married couples buying seats to a concert.

Three married couples have bought $6$ seats in a row for a concert. How many ways can they be seated if no man sits next to his wife. I have worked through this problem and have got the correct answer. The problem I am having is, I can’t, for the life of me, wrap my […]

The limiting case of a discrete probability problem

Say there are three jars, $j_1, j_2, j_3$ filled with different binary sequences of length two. The distribution of the binary sequences in each of the jars is given by the $p_i^k(1-p_i)^{n-k}$, where $p_i = \frac{i}{m + 1}$ where $m$ is the number of jars, $i$ is the jar index, $k $is number of 1$$’s […]

$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$

Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere. What I’ve done: By conditional Jensen $$E(X^2|\sigma(Y)) \geq E(X|\sigma(Y))^2 = Y^2 $$ a.e and thus $||X||_2 \geq ||Y||_2 $. Analogously $||Y||_2 \geq ||X||_2 $ and […]