expected value of exponential compound poisson process
Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$.How...
View ArticleHow can we maximize the expected number of runs in a shuffled deck of cards?
Take a standard deck of cards, with 52 cards, 26 red and 26 black. A run is a maximum contiguous block of cards, which has the same color.So,(𝑅,𝐵,𝑅,𝐵,...,𝑅,𝐵) has 52 runs.(𝑅,𝑅,𝑅,...,𝑅,𝐵,𝐵,𝐵,...,𝐵) has...
View Articleexpected value of Hermite polynomials
Lets say we have$$ f(x)=\sum_{r=0}^\infty \frac{E(H_{e_r}(X)) (-D)^r e^{-\frac{1}{2}x^2}}{r!}$$where $X$ is some symmetric random variable $H_{e_r}(X)$ is probabilist's Hermite polynomials and...
View ArticleWhat kind of entity is the expectation of a random variable?
Given a probability space $(Ω, \mathcal F, P)$ and an integrable random variable $ξ: Ω \to \Bbb R$, what kind of entity is the expectation of $ξ$ i.e. $E(ξ)=\int_ΩξdP$? My sense is that we have a...
View ArticleLaw of total variance on conditional expectation
Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then the law of total variance states that:$$ V[X] = V[E[X|Z]] + E[V[X|Z]] $$Then what if I...
View ArticleI have a n balls and m boxes problem [closed]
Consider the problem of distributing $n$ balls into $m$ boxes. Let $X_1, X_2, ..., X_{⌈log2n⌉}$ be a set of independent and uniformly distributed random variables with values in the range $\{1, 2, ...,...
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Sarah the squirrel problem (Linearity of Expectations)
From https://brilliant.org/wiki/linearity-of-expectation/I have two questions:First questionThe "obvious" solution is by using linearity of expectations. My idea was to write the states as $E(n -...
View ArticleTower law and iterated expectation
Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then applying the tower law in an iterative fashion we get:$$ E[X] = E_Y[E_{X|Y}[X|Y]] =...
View ArticleHow to differentiate $E(x)^2$?
How do you differentiate the function$$y_t=E(x_t)^2$$where $x_t = x_0+ar_t+c_t$, $E(c)=0$ and variance = $σ(u)^2$, and $x_0$ is a constant.I am unsure if my reasoning is correct in the method...
View ArticleVariance of expectation when updating from binary signals
Consider the following problem: There's a stream of signals $s=(s_1,...,s_T)$ with $s_t \in \{0,1\}$ and $P(s_t = 1) = p_t$. Now, $p_1\sim U[0,1]$ and, for every $t>1$, $p_t = p_{t-1}$ with...
View ArticleDerivation of variance and similar functions
I am interested in logical proof for the formula for variance:$$Var(x) = E((x-E(x))^2)$$The condition we are starting with is:$$Var(x+y)=Var(x)+Var(y)$$Here x and y are independent random variables.Of...
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Why is $P_{\rho}$ is a probability measure on the Borel subsets of $H\ $?
I am going through a paper on Operator Probability Theory by Stan Gudder. The author introduced the notion of probability distribution of self-adjoint operators on a Hilbert space where the...
View ArticleUnderstanding the recurrence relation of Matousek-Sharir-Welzl
I am unable to understand the following recurrence relation from the paper: https://people.inf.ethz.ch/emo/PublFiles/SubexLinProg_ALG16_96.pdf. Why is the probability for $t_k(n-1)$ not being accounted...
View ArticleExpected number of slots
Let's say we have an expanding system of slots(you can be keep increasing the number of slots). Now I have a very large number of coins in a bag, all equi-similar, labeled(1,2,..n). Now the experiment...
View ArticleBinary search expected number of comparisons?
What is the expected number of comparisons needed to find a number (if it exists) in an array of [1,2,...,n] using binary search? I understand the limiting time complexity of binary search is...
View ArticleConfusing expectations, sampling with and without replacement
This is an easy question about taking expectations that I'm confusing myself with. To describe the simplest setting, suppose that I have $N$ machines, which when chosen outputs a value $1$. I'll choose...
View ArticleExpected values with Maximum choice within integrals
I have a Manager who has no information about the profit from investment into two bonds, other than the fact that they are independently drawn from uniform distribution ${m, 1}$ where, m>0. Profit...
View Articlediscounted value of american option
Let $S(t)$ be the stock price at time $t\geq 0$ with $S(0)=s$ and $C$ an american option with payoff function $P(s)$. For $k>0$ let $\tau=\inf\{t:S(t)< k\}$ be the time where the option is...
View ArticleCutting a cake n-times - expected number of pieces
I'm trying to solve a brainteaser:A 2D circle (cake) is cut randomly with straight lines n-times. What's the expected number of pieces?Could you give me a hint on how to approach this? Is it possible...
View ArticleProve the next property of conditional expected value
Let X and Y be random variables with CONTINUOUS distributions. Prove that for every bounded function $g: \mathbb{R}^n \to \mathbb{R}^n$ , continuous almost everywhere, exist the next...
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