Trouble with an expression of expected value
So I am reading this proof about finite irreducible markov chains which says that states have unique positive stationary distribution $\pi$ such that $\pi_j=\frac{1}{\mu_j} \forall j$. The problem is...
View ArticleAbsolute Difference between two iid Censored Poisson Draws
Let $Y, Y' \sim Poisson \left( \lambda, C \right)$, where $\lambda$ is the mean and $C$ is some upper censoring limit. In this censored Poisson distribution, the probability mass beyond $C$ in an...
View Articleexpected value over all subsets of a set
We define the subweighted value $x$ of an ordered set $A$ with $k$ elements if for each element $a_i$ in $A$ we alternate the sum of its indexed weight; that is $x = \sum_{i=1}^{k} ((-1)^{i+1} \cdot i...
View ArticleHow do you calculate the average wagering of a casino offer?
I'm trying to calculate the expected value of casino offers that involve wagering. Let's say I have a £50 bonus that has to be wagered 40x (£2000 total) on a slot with an RTP of 97%. It says online...
View ArticleMarginalization and conditioning with expected values
I may have missed this during my intro stats/prob course but what is the difference between:$E_Y[X]$ and $E[X|Y]$?It seems like one you are marginalizating over and the other you are conditioning on. I...
View ArticleConditional Expectation, Conditional Probability [closed]
We have two random variables, X and Y, representing the market valuations of a product in two different regions. The probability density functions (PDFs) and cumulative distribution functions (CDFs)...
View ArticleExpectation, Joint PDF, Joint distribution
X is a random variable with PDF f(x) and CDF F(x). Y is a random variable with PDF f(y) and CDF F(y). Z (x,y)=X if x>y and Z(x,y)=Y if y>x. Determine E[Z] (expectation of Z).
View ArticleExpectation of a random variable and its reciprocal
Consider two RV $X$ and $Y$ which are both strictly positive.If $E(X)>E(Y)$, does this also imply that $E(1/X)<E(1/Y)$?
View ArticleCalculating the expected difference in observations
Let $X_1,X_2$ be i.i.d $N(0,\sigma^2)$. My tasks is to calculate the expected value of the function $T=|X_1-X_2|$. Intuitively, I want to say that this value would equal the standard deviation, but I...
View Article$\mathbb{E}[f(T)]$ as an integral over $E$ and $\Omega$ with change of variables
I am working on the following exercise:Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let $(E, \mathcal{E})$ and $(F, \mathcal{F})$ be measurable spaces. Let$$ Y : (\Omega,...
View ArticleExpectation and Conditional Probability
An urn contains $a$ white and $b$ black balls, where $a$ and $b$ are positive integers. One ball at a time is randomly drawn until the first white ball is drawn. find the expected number of black balls...
View ArticleDoes the Kelly Criterion assume that all future bets will stay the same?
In most explorations of the Kelly Criterion I’ve seen, we’re deciding the % of our bankroll to apply to a bet under the assumption we will be repeatedly faced with the same bet many times.I’m curious...
View ArticleInfinite Second Moment implies Growth condition on Expected Maximum
Let $X, X_i$, $1 \leq i \leq n$ be IID random variables such that:$$\mathbb{E}X^2 = +\infty$$I am trying to show that this implies a growth condition on$$\mathbb{E}[\max_{1\leq i \leq...
View ArticleUsing the symmetry of the normal to find $E(Z|Y)$ given $Z\sim N(0,1)$ and...
I know that this question has been answered before; namely here.I'm really struggling to understand the formulation of how this answer was reached. Specifically, as someone pointed out in the comments,...
View ArticleFinding $\mathbb{E}(T)$ from a martingale
Let $X_i, i=1,2, \ldots$ be i.i.d. random variables with $\mathbb{P}\left\{X_i=1\right\}=p=1-\mathbb{P}\left\{X_i=-1\right\}>\frac{1}{2}$, $S_0=0$, and $S_n=\sum_{i=1}^n X_i$ for $n \geq 1$. In...
View ArticleConjecture: If $x_k$ are random in $(0,\pi/2)$ then expectation of...
Let $E(n)=\text{expectation of }\dfrac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$ where $x_k$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.Is the following...
View ArticleUsing the Martingale Representation Theorem to generalize the Geometric...
I'm having trouble understanding the reasoning given in a proof of the following theorem in the book A First Course in Statistic Calculus by Louis-Pierre Arguin.Theorem 7.26. Let $(M_t, t \in [0, T])$...
View ArticleRegion of feasible $(E,V)$ for a discrete random variable supported on...
Given a discrete random variable taking on the first $k$ positive integer values with probability $p_i$, $i=1,\dots,k$, how can I compute the region of feasible combinations $(E,V)$, where...
View ArticleExpectation value of the Frobenius norm of a matrix
Let $A \in R^{n×n}$ and $B \in R^{n×n}$ be two matrices. Let $s$ be some positive integer. Let $\tilde{A} \in R^{n×n}$ be a random matrix with mutually independent entries:$$ \tilde{a}_{ij}...
View ArticleUpon Inspection, Expected Poisson interarrival time is $2/\lambda$?
Poisson distributions are memoryless, in particular:$$P(X_j>t+s|X_j>s)=P(X_j>t)=e^{-\lambda t}$$ where $X_j$ is the interarrival time. In particular, this implies that $E[X_j|X_j>s] =...
View Article