Computing expectation for partial collection in Coupon's Collector problem
I'm trying to code the formula below in numpy (quad), and I couldn't determine what $u$ is. The formula from Flajolet et. al gives the expectation for partial collection of size $j$ for the coupon's...
View ArticleFair value of coin tossing game
A biased coin is flipped $100$ times. The head and tail probabilities are unknown and the probability of heads is uniformly distributed between $0\%$ and $100\%$. if you guess correctly, you get $\$1$;...
View ArticleComputing the variance of a test statistic obtained from 2 samples.
Given that I have two samples $S_X$ and $S_Y$, where the samples of $S_X$ are drawn from $P_X$ and the samples of $S_Y$ are drawn from $P_Y$. The test statistic between the two samples are created in...
View ArticleRescaling of random variables
Say I want to study a function on a space of probability distributions, say for simplicity $g(X/\mathbb{E}(X))$ where $X$ is positive and runs in $L^1(\Omega)$-random variables with positive mean....
View ArticleDefining a unique, satisfying expected value from chosen sequences of bounded...
Let $n\in\mathbb{N}$ and suppose function $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$, where $A$ and $f$ are Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, where...
View ArticleExpected value of a game
Suppose you are at a casino. You are playing a game where your goal is to maximize your return from the game. The game is defined by the following rules:$\bullet$ You pay $1$ dollar to play the...
View ArticleWhen to Stop Pulling Balls from an Urn: What's the right way to solve?
We are pulling balls from an urn. Suppose that there are $n$ balls, of which $g$ balls are green. We can pull as many balls as we want, and for every non-green ball pulled, we get $1. If we ever pull a...
View ArticleA simple probability question: find expected number ... [closed]
Suppose there are n students in class, and they each complete an assignment. We hand back assignments randomly. What is the expected number of students that receive the correct assignment? When n = 3?...
View ArticleFisher information of normal distribution with unknown mean and variance?
I am asked to find the fisher information contained in $X_1 \sim N(\theta_1, \theta_2)$ (ie: two unknown parameters, only one observation). How would I find the Fisher information here?I know that with...
View ArticleConfusion over calculation of expected value of sample variance
I want to show that the sample variance $$S^2 := \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X_n})^2$$ is unbiased, i.e. $\mathbb{E}[S^2] = \sigma^2$.I know that the fastest way of showing this is by adding...
View ArticleCondition for Probabilistic Integrability?
$X$ is a compact subset of $\mathbb{R}$. $C(X)$ is the set of all continuous functions on $X$. $B(X)$ is the Borel sets of $X$. $\mu$ is a probability measure on $B(X)$. Is$$\int_X fd\mu$$ well-defined...
View ArticleVariance in the number of triangles in a random graph of size n
Consider the set $V=\{1,2,…,n\}$ and let $p$ be a real number with $0<p<1$. We construct a graph $G=(V,E)$ with vertex set $V$, whose edge set $E$ is determined by the following random process:...
View ArticleThe probability of a number is an element of a set [closed]
Alice and Bob agree on a interval $I=[1,10]$ with size $|I|=10$.Alice and Bob agree on a quantity, let's say $q=3$.Alice draws $q$ random unique numbers in the interval $I$ composing the set $C$,...
View ArticleExpectation of the ratio of two i.d. variables
Let $X, Y$ be two identically distributed (i.d.) positive random variables. If they are furthermore independent, from the Cauchy-Schwarz inequality (last step), one has$$ E[X/Y] = E[X].E[1/Y] =...
View ArticleCorrelation between IID Gaussian matrix and its Moore-Penrose inverse
$\bf H$ is $N$ x $K$ ($N$> $K$) random matrix that each component follows i.i.d. complex Gaussian with zero mean and unit variance.${\bf A} = {\bf H}^{\dagger}$ is Moore-Penrose inverse of $\bf H$,...
View ArticleOptimal Number For An Auction Bidding Hack
Four 50-sided dice are rolled so that the numbers are all different, then assigned randomly to players A,B,C,D. The players with the highest/lowest numbers pair up, as do the two middle ones. Then the...
View ArticleFind the expected value of the imbalance in this league [closed]
A baseball league has 6 teams. To decide the schedule for the league, for each pair of teams, a coin is flipped. If it lands head, they will play a game this season, in which one team wins and one team...
View ArticleExpectation of ratio of two identically distributed random variables is...
Let $X, Y$ be two identically distributed (i.d.) positive random variables. If they are furthermore independent, from the Cauchy-Schwarz inequality (last step), one has$$ E[X/Y] = E[X]\cdot E[1/Y] =...
View ArticleClarence Bread Quant question
Clarence is getting this bread. Literally. Clarence has $7$ small loaves and $3$ large loaves of bread in a bag. He draws in his bag and pulls out a loaf of bread uniformly at random, one-by-one. If it...
View ArticleExpectation of a multivariate Gaussian after going through a Softmax
Let $\varepsilon\sim N(0, I_D)$ be a $D$-dimensional random vector, distributed normally with mean $0$ and covariance given by the identity matrix of size $D$. In some computations I'm doing in my...
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