Articles of probability

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 […]

Littlewood's Inequality

Working on an exercise from Shorack’s Probability for Statisticians, Ex 3.4.3 (Littlewood’s inequality). Can’t seem to find any material for this. Help is appreciated. Prove:Exercise 4.3 (Littlewood’s inequality) Let $m_r \equiv \mathbb{E}|X|^r$ denote the $r$th absolute moment. Then for $r \ge s \ge t \ge 0$, we have $m^{r-s}_t m^{s-t}_r \ge m^{r-t}_s$. Thank you very […]

n people & n hats: probability that at least 1 person has his own hat

Suppose n people take n hats at random. What is the probability that at least 1 person has his own hat? The proposed solution uses inclusion-exclusion principle and gives the answer: $$\sum^{n}_{r=1} (-1)^{r-1} \frac{1}{r!}$$ But I think there is simpler solution: let’s calculate the complement, i.e. no one gets his own hat. Total number of […]

Can any function of the second moment of a random variable be recovered from its quantile function?

Summary of question It is known that the expected value of a random variable can be obtained from integrating its survival function. This is easily restated in terms of the quantile function as: $$ \int_0^\infty S(x)\;dx = \int_0^1F^{-1}(x) $$ whose equivalence can be seen graphically as integrating over the same area. This is useful, as […]

Calculating probabilities over longer period of time

There’s a great question/answer at: Calculating probabilities over different time intervals This is an awesome answer, but I’d like to ask a related question: What if the period goes the other direction, for example, the probability is determined for a year, but you want to see the probability of it happening over 50 years? For […]