Estimating the expected number of steps in the process
I share here the problem I saw on a forum.Problem:We have 7 batches of 8 small squares, each batch having its own color. We randomly assemble these small squares 2 by 2 to form 28 dominos. We remove...
View ArticleExpected win ratio in a game with an unknown and resettable win probability
Here's a simple one-player game:In each round of the game you can either win with probability $P$ or lose with probability $1-P$. $P$ is drawn randomly from the uniform distribution $U(0,1)$, its value...
View ArticleExpected Decay in the Number of Black Balls Over Iterative Sampling
Problem StatementWe begin with $N$ black balls. At each iteration, denoted by $\mathcal{D}$ steps in total, we perform the following operation: during the $i$-th step, we introduce $k^{i}N$ white balls...
View ArticleOn average, how many times must I roll a dice until I get a $6$?
On average, how many times must I roll a dice until I get a $6$?I got this question from a book called Fifty Challenging Problems in Probability. The answer is $6$, and I understand the solution the...
View ArticleWhy is independence needed in this simple conditional expectation problem?
Suppose that $X$, $Y$, and $Z$ are random variables on some probability space. Suppose $Z$ is indpendent from $X$ and $Y$. Prove that:$$E(X|Y, Z) = E(X|Y)\quad a.e.$$As the RHS is $Y$-measurable, it is...
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WLLN for IID random variables
Examine if WLLN holds for the sequence $\{X_n\}$ of IID random variables with$$P\bigg(X_i=(-1)^{k-1}\cdot k\bigg)=\frac6{\pi^2k^2};\ \ \ k=1,2,3,\dots;\ \ \ i=1,2,3,\dots$$Above is an example in my...
View ArticleRescaling of random variables
Say I want to study a function on a space of probability distributions, say for simplicity $g(X/\mathbb{E}(X))$ where $X$ is positive and runs in $L^1(\Omega)$-random variables with positive mean....
View ArticleDifferentiate the Normalization Term of the Binomial to Get the Expectation
I am working through the problems in Bishop's Pattern Recognition and Machine Learning; I am currently on exercise 2.4 The problem asks us to show that the mean of the binomial distribution by...
View ArticleCan we refine the paper "Integration with Filters" by Bottazzi E. and Eskew...
Motivation:In a magazine article on problems and progress in quantum field theory, Wood writes of Feynman path integrals, “No known mathematical procedure can meaningfully average an infinite number of...
View ArticleLet $X^2 \sim \chi^2_n$ is there an easy explanation as to why...
Let $X^2 \sim \chi^2_n$. One can compute that$$\mathbb{E}[X(X^2 -n)] = \mathbb{E}[X] = \sqrt{2} \frac{\Gamma((n+1)/2)}{\Gamma(n/2)}.$$However is there an "intuitive" explanation as to why these two...
View ArticleSimple(r) way to derive the expectation of an inverse Wishart?
I am looking for a simple way to derive the expectation of an inverse Wishart matrix., with distribution $W^{-1}$ where $W=\sum_{i=1}^n \Sigma^{1/2} g_i g_i^T \Sigma^{1/2}$ for a covariance $\Sigma\in...
View ArticleDoes this Approximation exist for the following relation: $𝐸[ (𝐴−𝐵)^2 ] ≈𝐸 [...
Let $A$ and $B$ are independent RVS (positive real). Is there a relation to approximate the expectation of the squared difference, like this: $𝐸[ (𝐴−𝐵)^2 ] ≈𝐸 [ A^2 ]$−$𝐸 [ 𝐵^2]$. This seems to rely on...
View ArticleBasketball Betting Pool - Expected Number of Points...
There are 64 teams who play single elimination tournament, hence 6 rounds, and you have to predict all the winners in all 63 games. Your score is then computed as follows: 32 points for correctly...
View ArticleBounds on restricted expectation
Are there any useful bounds on the restricted expectation $f(x)=\mathbb{E}[X1_{X>x}]$ for a positive random variable with finite expectation, but not bounded? Since the probability of the...
View ArticleLaw of Total Expectation and Second Moment [closed]
Is this equation true?$$E(X^2|X \gt 1)=E(X^2)-E(X^2|X \le 1)$$I am computing $\operatorname{Var}(X|X \gt 1)$ but getting a wrong result. I suspect this could be where I am wrong.
View ArticleImproper Integral for expected value of a random variable
Let $X: \Omega \to [0, \infty)$ be a random variable. Let $F_X$ be its distribution function and $f_X$ its density function. By definition$$\mathbb{P}[a \le X \le b] = F_X(b) - F_X(a) = \int_a^b...
View ArticleFirst hitting time of a nonhomogeneous Poisson process $N(t)$ over a general...
I find this question when doing my own research, which relates to the first hitting time of a nonhomogeneous Poisson process $N(t)$ over a general function $f(t)$. I want to know the pdf or the...
View ArticleLet $T$ be the number of tosses required until three consecutive heads appear...
A fair coin is tossed repeatedly. Let $A_{n}$ be the event that three heads have appeared in consecutive tosses for the first time on the $n$-th toss. Let $T$ be the number of tosses required until...
View ArticleAnalytical methods to derive expected points from $\texttt{xG}$ differential...
In football, Expected Goals ($\texttt{xG}$) quantifies the likelihood of scoring from shots, where a team’s $\texttt{xG}= k$ is the sum of probabilities $\left\{ p_{1}, \ldots, p_{N} \right\}$ of $N$...
View ArticleIf $X_n \xrightarrow{\text{a.s.}} X$ and $\left\lvert f(x)-f(y)\right\rvert...
Let $X_1,X_2,\dots$ be a sequence of real random variables such that$$X_n \xrightarrow{\text{a.s.}} X$$as $n\to \infty$. Let $f:\mathbb R \to \mathbb R$ be a function that satisfies$$\left\lvert...
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