How to calculate probability of bankruptcy for $E>0$ and given $sigma$?
I will demonstrate the problem with the simple example. Suppose I want to start a casino. The probability theory tells me that the expected value for my games is greater than zero ($E>0$), and...
View ArticleDegeneracy paradox
Say I have a highly biased coin that lands heads with $p_h=0.01$ and tails with $p_t=0.99$, and I flip it $98$ times. The probability of zero heads is ${p_t}^{98} approx 0.373$. The probability of one...
View ArticleExample Bayesian resolution of the Two Envelopes Problem [closed]
What is a concrete example of a Bayesian resolution to the Two Envelopes Problem?
View ArticleIn which case $mathbb E[X]=sum _ix_i P[x_i]$ can be $0$ when all $x$'s are...
Say $X$ is a random variable and $x$’s are realizations of $X$ . Say , $mathbb E[X]=sum _ix_i P[x_i]=0$ . But I do not understand in which case $mathbb E[X]=sum _ix_i P[x_i]$ can be $0$ when all $x$’s...
View ArticleWhy is $mathbb E(X)=sum_{i=1}^{n}x_i P(x_i)$?
If $X$ is a random variable and $x$’s are the realizations form $X$ and $N$ is the population size $n$ is the sample size Which one is correct $mathbb E(X)=sum_{i=1}^{N}x_i P(x_i)$ or $mathbb...
View ArticleHow to compute $P(|X – E_Y[h(y)]|< c)$?
Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X – E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and...
View ArticleExpectation of $b^T operatorname{sign}(Ab)$
I’m trying to compute the expectation of: $$b^T operatorname{sign}(Ab)$$ Where $b$ is a $ntimes1$ vector of independent Bernoulli random variables: $$P(b_i = 1) = 0.5,quad P(b_i = -1) = 0.5$$ and $A$...
View ArticleExpected value of a diagonal
If I have $E[A] = B$, where $E$ is the expected value, $A$ and $B$ are square matrices and $text{diag}$ refers to the vector of coefficients on its diagonal. In this case, what will be the value of...
View ArticleExpectation over training and testing set
If I have a linear regression model with $p$ parameters that fit by least squares a training set $(x_1, y_1),…,(x_n, y_n)$ drawn at random from a population, and we have some test data $(tilde{x}_1,...
View ArticleFind the expected frequency of some state in a state sequence of length N...
I can represent stochastically-articulated sequences of states using a transition matrix M where a given entry in cell (i,j) corresponds to the probability of state j given that the current (or, most...
View Articleconditional/unconditional expectation and variance for an AR(1) process
We have an AR(1) process, $X_t=alpha X_{t-1}+varepsilon_t$ with $varepsilonsim(0,sigma^2)$, $X_0=0$ and $|phi|<1$. We have the conditional expected a value with respect to $X_{t-1}$: $E(X_t|X_{t-1})...
View ArticleMoments of linear combinations of normally distributed variables
So I have this problem I have to solve for my homework for my Linear Regression Class that is due tomorrow and I honestly dont know where to start with this problem. “Suppose to have $Xsim N(0,1)$...
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Negative Sampling Expectation in word2vec algorithm
My maths is rusty and I’ve been reading up on word2vec and am not sure whether I understand the expected value in the objective function. It can be found in equation (4) in one of Mikolov’s original...
View Articlefinding variance and expected value – multivariate case
I would like to ask you a question – I bumped into a problem that I do not know how to solve- Let $X_1;dots;X_n$ and $Y_1;dots; Y_m$, be two random samples from distributions with means $mu_1$ and...
View ArticleExpectation of a chi-squared distribution
I am to find out the value of this expectation : $$E bigg(frac{U^p}{U+V} bigg),$$ where U $sim$ $chi^2_1$ and V $sim$ $chi^2_n$. U and V are independent. Can anyone give me any hints about how to start...
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Determine the best sample size for minimum expected loss
Let $theta sim Gamma(1,2)$ and $X_1,…,X_n$ iid such that $X_i|theta sim Poisson(theta)$. It is asked to determine the best sample size $n^*$ such that the posteriori risk $$L(theta, d) = (theta-d)^2 +...
View ArticleMaximise expectation of exponential given mean and variance
The problem is as follows: Suppose that $X$ is a random variable with $mathbb{E}X=0$, $mathbb{D}X = sigma^2$ and having a finite support: $P(|X|leq a)=1$. What is the maximum possible value of...
View ArticleValue of $mathbb{E}(frac{1}{aX+b})$?
There are rules e.g. $mathbb{E}(aX+bY)=amathbb{E}(X)+bmathbb{E}(Y)$, where $X$, $Y$ are random variables and $a$ and $b$ are constants. But what about $mathbb{E}(frac{1}{aX+b})$?
View ArticlePopulation vs sample
When we think of linear regression, the implicit assumption is that we only observe a small fraction of a possibly infinite large population. Thinking of simple averages, imagine a fair die. The...
View ArticleExpected value of a marginal distribution when the joint distribution is given
I am asked to find the expected value of a vector of two random variables when the joint density is given. Is the recipe for solving this problem: Find the marginal distributions Find the expected...
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