Number of moves necessary to solve Rubik's cube by pure chance
Suppose, random moves are made to solve Rubik's cube. A move consists ofa $90$-degree-rotation of some side. The starting position is also random.What is $E(X)$, where $X$ is the number of moves until...
View ArticleFinding probablities (to find the expected value) in a game of throwing a...
The question I saw was: In a game, a man wins 100 points if he gets 5 or 6 on a throw of a fair die and lose 50 points for getting any other number. If he decides to throw the die either till he gets a...
View ArticleConditional Expectation when variables are drawn from uniform distribution of...
I have the following problem -$a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $\{m, 1+m\}$ and $b$ is drawn from uniform distribution $\{0,1\}$....
View ArticleSpecial case of expected value convergence
Problem:I have a special case of convergence of random variables and I would like to show that their expected values also converge.I have a sequence of random variables $\{X_t\}$, $X_t...
View ArticleFinding conditional expectation when variables are drawn from uniform...
I have the following problem -$a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $(m, 1+m)$ and $b$ is drawn from uniform distribution $(0,1) $....
View ArticleDefinition of Covariance operator.
Let $x\in H$ for some Hilbertspace $H$. Recall the Covariance operator $\mathbb{E}[x\otimes x]$, where $x \otimes x := \langle x, .\rangle x$. How is actually the expectation defined in this case. For...
View ArticleConvergence of expected values under the mapping
I have a sequence of random variables $X_k \colon \Omega \to [0, 1]$, $X_k \stackrel{a.s.}{\to} X$, where $X \colon \Omega \to [0, 1]$. I know that $\mathbb{E}X_k \to \mathbb{E}X$. I want to prove that...
View ArticleProve lower bound on probability of ball remaining indefinitely
The problem is that we are given a magical ball. Every second (starting from $t = 0$), it either disappears, duplicates, or triplicates with equal probability. All balls are independent from each...
View ArticleWhen trying to guess in which trial a dice will show the number 4, should we...
According to the geometric distribution the probability of getting a 4 when a dice is thrown goes down with each throw...
View ArticleThe probability that all magic balls will eventually disappear: A branching...
The problem is that we are given a magical ball. Every second (starting from $t = 0$), it either disappears, duplicates, or triplicates with equal probability. All balls are independent from each...
View ArticleWhen is an expectation bounded?
I would very much appreciate some input on the following.Assume I have a random variable $v$ on a probability space $(X,F,P)$ and function of the random variable, $f(v)$.Is the statement$$P(\{x\in X...
View ArticleMaximizing an expression which involved $p$ which probability of a vaccine...
The QuestionA particular Covid-19 vaccine produces a side effect (fever) with probability $p \in[0,1]$. A clinic vaccinates 500 people each day and records the number of people having side effects on 5...
View ArticleEqualizing Expected Values by transforming RV
Let us consider a function $f(\cdot)$, that can be assumed to be non-negative. Let us also consider two $n$-dimensional random variables, $X_0$ and $X_1$, with probability distributions...
View ArticleHow do we know whether $X_{i}$'s are independent and what is the specific...
The problem is given as :-For a continuous real valued function $\Large f$, defined on $\Large [0,1]$ calculate :- $\Large \underset{n \to \infty}{\lim} \underset{0}{\overset{1}{\int}} \cdots...
View ArticleConditional expectation of two exponential random variables
$X$ and $Y$ are two independent exponential random variables:$P(X \geq x)=e^{-\lambda_1 x}$ for every $x>0$;$P(Y \geq y)=e^{-\lambda_2 y}$ for every $y>0$;How to calculate the expectation of $X$...
View ArticleFisher information of poisson distributed random variable
Let's consider a printer queue. We know that the expected number of printer jobs almost obeys a Poisson distribution, so $P_{\vartheta}(X=k)=e^{-\vartheta}\frac{\vartheta^k}{k!}$, where...
View ArticleEquivalent condition for Lindeberg-Levy-Feller Central Limit Theorem.
Let $\left\{X_{n}\right\}_{n \geq 1}$ be a sequence of random variables. Let$$S_{n}=\sum_{j=1}^{n} X_{j}, \quad s_{n}^{2}=\sum_{j=1}^{n} \mathbb E\left(X_{j}^{2}\right)<\infty$$If $s_{n}^{2}...
View ArticleHow to derive the expected number of rolls until a number appears $k$...
In Expected number of rolls until a number appears $k$ consecutive times, the formula was given to be $E[N] = \frac{6^N - 1}{5}$. You can prove this formula using induction like in the accepted...
View ArticleStrange relationship between two expectations related to die rolls
Let $X_n$ be the expected number of die rolls to get $n$ consecutive sixes. $\qquad\qquad\qquad\qquad\qquad$Let $Y_n$ be the expected number of die rolls to get any number$n$ times consecutively.$X_n$...
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Expectation of $u^\top(u + Ax)$, when $A$ and $u$ are nonlinear functions of $x$
Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that$$u = s-y \\v =(\operatorname{diag}(s) - ss^\top)x$$I am interested in the inequality $u^\top (u +...
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