Articles of probability

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 […]

Probability Bayesian network problem

The diagram above is the Bayesian network of my problem. I want to find $$\Pr(B=F \mid E=F, A=T)$$ I have evaluated it into the following steps, then I got a bit stuck: $$\Pr(B=F \mid E=F, A=T) = \frac{\Pr(B=F,E=F,A=T)}{\Pr(E=F,A=T)}$$ $$=\frac{\Pr(B=F,E=F,A=T)}{\Pr(A=T\mid E=F)\times\Pr(E=F)}$$ $$$$ I was able to get $\Pr(B=F,E=F,A=T)$ from: $\Pr(B=F,E=F,A=T)=\Pr(A=T\mid B=F,E=F) \times \Pr(B=F,E=F)$ $\Pr(B=F,E=F,A=T)=\Pr(A=T\mid B=F,E=F) \times […]

Moments and non-negative random variables?

I want to prove that for non-negative random variables with distribution F: $$E(X^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$ Is the following proof correct? $$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x)) dx$$ using integration by parts: $$R.H.S = [x^{n}(1-F(x))]_0^\infty + \int_0^\infty x^{n} f(x) dx = 0 + \int_0^\infty x^{n} f(x) […]

2011 AIME Problem 12, probability round table

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. I have been searching around math stackexchange […]

Connection between the Kalman filter and the multivariate normal distribution

Consider at dynamic linear model where $$ \theta_{1} \sim N(\mu_{1}, W_{1}), $$ $$ \theta_{i}=G\theta_{i-1} + w_{i}, w_{i}\sim N(0,W), $$ $$ Y_{i} = F\theta_{i} + v_{i}, v_{i}\sim N(0,V) $$ and $ \theta_{1}, w_{i}, v_{i} $ all independent random vectors. Let $ \theta_{0:t} : = (\theta_{t}, \theta_{t-1},\ldots, \theta_{0}) $ and $ Y_{1:t}:= (Y_{t},Y_{t-1},\ldots, Y_{1})$. A generel result […]

Extension of the Birthday Problem

How do you find the expected number of people (or the expected number of pairs) among the n that share their birthday within r days of each other? For the regular birthday problem, it’s $n\left(1-(1-1/N)^{n-1}\right)$ expected people or ${n\left(1-(1-1/N)^{n-1}\right) \choose 2}$ pairs (see https://math.stackexchange.com/a/35798/39038). In this link, is it correct to derive the expected number […]

Sum of Cauchy distributed random variables

Problem: Let $X_1, X_2, \ldots $ be independent $C(0,1)$ and set $S_n = \sum_{k=1}^n X_k$. Show that $\frac{1}{n}\sum_{k=1}^n \frac{S_k}{k}\sim C(0,1)$. Using the characteristic function it is easy to get that $\frac{S_k}{k}$ is $C(0,1)$. But $Y_k=\frac{S_k}{k}$ are not independent for different $k$ so that cannot be applied directly in this case.