Circular random walk - a biased markovian frog
Inspired by a recent question I tried to generalize the problem and has considered a biased frog which jumps clockwise (left) and counterclockwise (right) with different probabilities $q$ and...
View ArticleExpectation of a Poisson random variable
Let $N_{\lambda}$ be a random variable having Poisson distribution with probability mass function given by$$ P( N_{\lambda} = k) = \frac { \lambda^k e^{- \lambda} }{ k!}, $$ where $\lambda >0.$Now,...
View ArticleCalculating Expected Value of 1/(S+1) for a Poisson Binomial Random Variable
Let $S$ be a Poisson Binomial random variable, i.e. the sum of $n$ independent Bernoulli random variables with parameters $p_1, ..., p_n$. I'd like to know the expected value of $\frac{1}{1+S}$,...
View ArticleRepresenting a conditional expectation
Suppose I have a general random X, that is not necessarily continuous. It makes sense to me that I should be able to write:$$E(X|X>c)=\dfrac{E(X\cdot 1_{X>c})}{P(X>c)}$$If I assume that the...
View Article$E(XY)=E(X)E(Y)$ implies $X\perp Y$ when $X$ and $Y$ take only two values
I have the next problem:Problem.Let $X$ and $Y$ be two random variables each taking only two values. If $E(XY)=E(X)E(Y)$, then $X$ and $Y$ are independent.I have tried to get a proof but I haven't be...
View ArticleThree players take turns throwing 3 dice. The first to throw a total of 11...
Three persons, $A$, $B$, and $C$, take turns throwing three dice once. The first player to throwa total of 11 is awarded a prize of £200, and the game ends. What is the game’s expected value for each...
View ArticleWhat is the expectation of $R^2$ for for iid $y, x_1, ... x_k \sim N(0,1)$
$Y, X_1, ...., X_k$ are all iid $N(0,1)$ with $n$ samples.I can't make any progress on this problem... I don't even know an approximation but from simulation it seems to be $k/n$.What is the...
View ArticleProve $E(X) = a$ for a pdf symmetric about $a$
Show that if $X$ is a continuous random variable with p.d.f. $f(x)$ such that $f (a + x) = f (a − x)$, then $E(X) = a.$I did a change of variable and set $x=a+y$, but I'm having difficulty getting to...
View ArticleHow many rolls are sufficient to ensure, with probability 99%, that the sum...
I roll a pair of fair dice $n$ times, and calculate the sum of all $2n$ faces which come up:Suppose each roll of each die is independent of other rolls.How many rolls are sufficient to ensure, with...
View ArticleReference Request: Probability-and-Convexity Statement
I can prove the following statement, but it would be even better to have a reference I can cite. Does anyone know where this (or a generalization) is proven?Proposition. Let $0 \leq b \leq a$, and for...
View ArticleWhat is the maximum volume of a statistical moment on a closed domain?
The objective is to find the maximal volume under a moment that is defined on a hypercube.Take domain the p-dimensional hypercube $\Theta=[0,l]^p$ .The moment is bounded$$|E_{p(y|x)}[b(y)]| \leq \rho \...
View ArticleDerivation of a conditional expectation
Setup. I have 3 random variables $Z_1,Z_2,\epsilon\in\mathbb{R}$, where$$(Z_1,Z_2)\sim\mathcal{N}\left(\begin{bmatrix}0 \\0\end{bmatrix},\Sigma=\begin{bmatrix}\Sigma_{11},\Sigma_{22}...
View ArticleConditional Expectation of $X/Y$ given $X \leq 2Y$ for Independent Uniform...
Let $X$ and $Y$ be independent random variables, each uniformlydistributed on $(0,1)$. Find the conditional expectation$\mathbb{E}\left[\frac{X}{Y} \mid X \leq 2Y\right]$.I've tried to solve this...
View ArticleDifferentiation through integrals, tower of expectations - Proof validation...
I am interested in minimizing a loss function involving probability densities of kernel density estimation functions (KDEs). One of the terms I obtain is this one:$$\int _{X}\left(\int _{Z} p( z) K(...
View ArticleExpected value of this distribution
Let $a>0$ and let $X$ be a random variable with pdf $f(x)=\begin{cases}2\phi_\sigma(x)+\frac{\theta}{a}&\text{if $0\leq x \leq a$}\\0&\text{otherwise}\end{cases}$, where$\phi_\sigma(x)$ is...
View Articlei roll a 6 and 4 sided dice. on first roll its a 2. what is the expected...
this question was asked before here, and here. but with the first answer is 3.1 (from comments) and from the second one hte answer is 2.9i am confused, where is the right one?
View ArticleExpected number of original items after some replacements
The problem is this: There are always 6 items on a shelf. Each day, two random items are purchased and two new items are added in their places. Suppose there is an infinite supply of items. What is the...
View ArticleGamma Distribution Moments Derivation
I got the following gamma function:$$ \frac{B^{a}}{\Gamma(a)}x^{a-1} e^{-Bx}\ I_{[0,\infty)}(x) $$I would like to derive the raw moments and thus the variance but I know that I must have made a mistake...
View ArticleExpected number of bonus wins for a specific slot machine game
I am analyzing a slot machine game for an assignment I was given.There are two reels, one on the left and one on the right.The probability of winning a spin is $p_L = 1/9$ for the left reel and $p_R =...
View ArticleProbability when two events need to happen sequentially
I have a random variable $X \sim Geo(p)$. I have a second random variable $Y \sim Geo(p)$. Both RVs describe the flip of a weighted coin with probability $p$ to get heads.But the two random variables...
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