Game theory expected value
We play a game involving two players. Each player calls a number 1 or 2. If the sum of these numbers are odd (i.e. equal to 3), then player 1 gets 3 points and player 2 loses 3 points. If the sum of...
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Existence of $E(u)$; $E(u)=E(u^+-u^-)$
From chapter $IV.2$ of $$\textit{William Feller 'An Introduction to Probability Theory and Its Applications'; Vol.2}$$$E(u)=\int_{\mathcal R^r}u(x)F\{dx\}$I'm confused about the statement '$E(u)$...
View ArticleMax of sums of independent elements in two n-tuples
Let $X_{1}, ..., X_{n}, Y_{1}, ..., Y_{n}$ be independent random variables with finite expectation. Assume that for $1 \leq i \leq n$, it is the case that $X_{i}$ is equal in distribution to $Y_{i}$...
View ArticleMutual information expansion not justifiable
I have recently read a mutual information term,$I(X;Y,Z)=E_{p(X,Y,Z)}\big[\log\frac{ p(X|Y,Z)}{p(X)}\big]$.While this expansion does not make sense to me. Is it correct? My understanding (using...
View Articlemoments of a Geometric brownian motion using Ito's lemma
Can someone explain me what is wrong in my derivation of the formula for the moments of a GBM using Ito's lemma (I am not interested in other methods) ?\begin{equation}dX=\mu Xdt+\sigma...
View ArticleShow that $\sigma_n^2$ is an unbiased estimator for $\sigma^2$
Let $X_n$ be a sequence of i.i.d. random variables with $E(X_i)=\mu,Var(X_i)=\sigma^2$ and define $\mu_n:=\frac{1}{n} \sum_{i=1}^n X_i,\sigma_n^2:=\sum_{i=1}^n(X_i-\mu_n)^2.$I have already shown that...
View ArticleExpected value of squared random variable if expected value of random...
Let $E[X]=0$ them How is Expected value of $E[X^2]$ ? in my opinion is $0$ But I need confirmation and explanationIn general, if $ (\Omega,\Sigma,P) $ is a probability space and $ X: (\Omega,\Sigma)...
View ArticleSymmetry of i.i.d. continuous r.v.s problem, part 2
I have the following problem:Athletes compete one at a time at the high jump. Let $X_j$ be how high the $j^{th}$ jumper jumped, with $X_1, X_2, \dots$ i.i.d. with a continuous distribution. We say that...
View Articlepart 2: We roll a fair die until 3 number 4s are thrown. Determine the...
this question came from this questionthe only difference i want to change is, lets assume the game stops once I roll 3 dice in a row with number 4. how would the problem change?the solution that caught...
View ArticleExpected winning time in fair gambler's ruin problem using martingale
In fair gambler's ruin problem, we already knew that the expected time of winning is $$E(\tau_n|\tau_n<\tau_0)=\frac{n^2-k^2}{3}$$ where $k$ is how much money we have in the beginning and $\tau_i$...
View ArticleClarification regarding betting game and linearity of expectation
Here is the problem I'm dealing with: You have 3 blue and 3 red cards. These cards are mixed and placed face-down in a deck, ready to be turned over one-by-one. Before each card is turned, you are...
View ArticleCar parking - quant guide
Suppose that we have 2 cars parked in a line occupying spaces 1 and 2 of a parking lot. Spaces 3 and 4 are initially empty. Every minute, a car is considered eligible to move forward one space ifa) the...
View ArticleAn application of Chebyshev association inequality?
Let $X$ be a r.v and let $f \geq 0 $ be a nonincreasing function, $g$ be a nondecreasingreal-valued function. Suppose $h\geq 0$ is a function such that $h(X)$ has finite expectation with $E[h(X)f(X)]...
View ArticleFinding Expectation of a Uniform random variable from its moment generating...
From Taylor series, if I need to get the k-th moment, I need to find the k-th derivative of the Moment Generating function.If I have $X \sim \text{Uniform}(0,1)$, the MGF $M_{X}(s)$ is; $$M_{X}(s) =...
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Find $\mathbb{E}[\det(A-A^T)]$ if $a_{ij} \sim \text{Bernoulli}(p)$
Let $A$ be a random matrix, which elements take values $1$ and $0$ with probably $p$ and $(1-p)$ respectively. What is $\mathbb{E}[det(A-A^T)]$?When dimensionality of $A$ is odd, $det(A-A^T) = 0$, as...
View ArticleInequality of ratio of expectations
Given non-negative functions $f,g: \mathbb{R} \rightarrow \mathbb{R}_{+}$ with $f(x) \leq g(x)$. Consider the following two ratios:$$\frac{\mathbb{E}_{x\sim\mathcal{N}(\mu_1,...
View ArticleExpectation of max of sums of independent r.v.s
Let $\{ X_{1}, X_{2}, Y_{1}, Y_{2} \}$ be independent with $X_{i} =^{d} Y_{i}$ for $i \in \{ 1, 2\}$ and $\mathbb{E}(X_{1}), \mathbb{E}(X_{2}) < \infty$. Is it true...
View ArticleHelp with probability problem involving tourist and bear encounters
I am trying to solve the problem below but having a hard time doing it, any help appreciated (my attempt at the bottom). Specifically, I do not know how to solve (b) and I am unsure whether I got other...
View ArticleMonotonicity with expectation
I think the following is true but I cannot prove it.Let $Z_1, Z_2$ are two random variables defined on the same sample space $\Omega$. Suppose that $Z_1(\omega) < Z_2(\omega)$ for all $\omega\in...
View ArticleTrying to find the conditional expected value of X when X + Y = m, and X, Y...
If X and Y are independent binomial random variables with identical parameters n and p, calculate the conditional expected value of X given X+Y = m.The conditional pmf turned out to be a hypergeometric...
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