**1.075 expected-value questions.**

Okay so there's a game, where you have 1 percent chance of winning an alpha pack and if you fail you your chance goes up by 3 percent. How would you calculate the expected value? I've only taken an AP ...

I am having trouble a question in my probability course and was looking for some insight into it.
A digital clock sits next to your bed and in the mornings when you don't use an alarm you notice Y =...

The expectation of random sum of independent identically distributed variables is given either by the law of total expectation or by Wald's identity. Are these generalised to tackle the random sum of ...

Consider the following model with the usual OLS assumptions: $\epsilon_i$ are uncorrelated random variables with mean zero and constant variance $\sigma^2$.
$$y_i=\beta+ 2 \beta x_i+\epsilon_i$$
$(...

How do you solve the integral
$$E(L(\theta_A,\theta_B)) = \int_0^1\int_{\theta_B}^1(\theta_A - \theta_B)f(\theta_A)f(\theta_B)d\theta_Ad\theta_B$$
where $\theta_A \sim Beta(\alpha_1, \beta_1)$ and $\...

I am reading this article:
https://www.sciencedirect.com/science/article/pii/S0040580901915424?via%3Dihub
Where as far as I can make out they are taking the expected value of the difference of 2 ...

For independent random variables $\alpha$ and $\beta$, is there a closed form expression for
$\mathbb E \left[ \frac{\alpha}{\sqrt{\alpha^2 + \beta^2}} \right]$
in terms of the expected values and ...

The image below shows the process for calculating the Expected value of a squared random variable X. From line 3 to 4 the unit step function [u(x) - u(x-1)] is removed and a 3 is placed before (1-x^2)....

Let $X$ be a random variable, and let $f$ be a concave function.
Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$?
Jensen's inequality only gives an upper ...

Suppose that $X_1, X_2, \ldots X_n$ is a sequence of random variables on a set $S$, drawn independently according to the pdf $p : S \to [0,1]$.
Part I:
Given some $f : S \to \mathbb{R}$, I want to ...

So, for two random variables $A$ and $B$, I know that $\mathbf{E}[A]$, $\mathbf{E}(B)$, $Var(A)$ and $Var(B)$, I also know that $\mathbf{E}[A] = \mathbf{E}(B) + C$, I wonder if I can find a simple way ...

If the two random variable functions are non-independent, then it doesn't make sense why the equation holds true. For example, let's create a table:
\begin{align}
f(x,y) & \\[5pt]
f(0,0) &= 0....

I was doing some work in scipy and a conversation came up w/a member of the core scipy group whether a non-negative discrete random variable can have a undefined moment. I think he is correct but do ...

I am a beginner in statistics and probability study. Today I came across a new topic called the Mathematical Expectation. The book I am following says, expectation is the arithmetic mean of random ...

I have a question that gives me the density of a Pearson type VI distribution and then says to state the range of parameter(s) for which the expression for the mean is valid. In your calculations, ...

This question came up during my research. In detail, let $X$ be Poisson distribution, i.e. $X\sim Pois(\lambda)$. I am interested in the expected value
$$ \mathbb{E}\left[\log\left(\frac{\Gamma(X+c)}{...

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the solution:...

If $x$ and $y$ are independent and normally distributed:$$x\sim N(\mu_x,\sigma_x)$$
$$y\sim N(\mu_y,\sigma_y)$$
and $r$ is a random variable with the following relationship to $x$ and $y$
$$r = \sqrt{...

Suppose $X\sim \mathcal{N}_d(\boldsymbol{0},\boldsymbol{\Sigma})$ follows a $d$-dimensional multivariate normal distribution. Let $\text{A}_i$ and $\text{A}_j$ be two arbitrary symmetric $d\times d$ ...

If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread.
Does this property have a name?
Are there ...

Is there any nice geometric interpretation of the mathematical expectation of a random variable (preferably based on density or cumulative density plot)?
(For example, median has a nice geometric ...

I'm a begginer in statistics.
I came across a problem, please help me out.
Is expectation always mean arithmetic mean of a variable?

I am revisiting a question I asked previously with a slight caveat. In my new situation, I am considering the marbles to always be attached to the same neighbors. Hopefully this will be clearer with a ...

What is the significance of using conditional expectation E(Y|X=x) in regression and not just use the regular expectation E(y). What will be the consequence if we use E(Y) instead? ( can we do this ...

Let $X\geq0$, $\eta\geq0$ and $X,\eta$ independent. We measure $X$ with a one-sided error: $\widetilde{X} = X - \eta$.
Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?

The Inverse Gaussian Distribution density is : $$\frac{\phi^{\frac{1}{2}}}{\sqrt{2\pi y^3}} exp[\frac{-\phi(y - \mu)^2}{2\mu^2y}]$$
Got to this integral: $$\int_0^\infty \frac{1}{y} \frac{\phi^{\frac{...

This is a proof of the per-decision importance sampling (theorem 1) from the appendix of:
https://www.google.co.uk/url?sa=t&source=web&rct=j&url=http://scholarworks.umass.edu/cgi/...

Consider a random vector $\boldsymbol{x} \in \mathbb{R}^N$ and the identity matrix $\boldsymbol{I}_N \in \mathbb{R}^{N\times N}$.
I have to compute the expected value of the following Kronecker ...

this is my first question here :)
Problem Statement
Let $h \in \mathcal{H}$ be a hypothesis to some class of binary classifiers $\mathcal{H}$. Show that
$$\mathbb{E}_{\mathcal{D}_n}\left[R_e(h)\...

I was wondering where there is a general formula to relate the expected value of a continuous random variable as a function of the quantiles of the same r.v.
The expected value of r.v. $X$ is defined ...

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 ...

I'm trying to understand the basic statistics involved in trading.
Suppose I'm trying to decide whether to buy a stock whose current price is $V_0$. Suppose I have some fancy statistical model from ...

You have a random variable $X$, which can take a value from $i = \{1,2,3,...\}$, where the probabilities are $p_i = 0.5^i$. Now you if we have a list $(x_1, ..., x_n)$ of independent observations of ...

This is a derivation I found online for the mean of a random variable:
Let $X_1, X_2, \dots, X_n$ be $n$ independently drawn observations from a
distribution with mean $\mu.$
Let $\bar X$ be the ...

I'm trying to determine the average life expectancy from the following model. I feel I'm close, but I can't quite get the right answer.
The daily death chance for any given individual is modeled as:
...

In Machine Learning classic models like MLP, Logistic Regression or Linear Regression are called discriminative models.
I frequently read that those models estimate the conditional probability ...

For a binary model with Y as the dependent variable and X1, X2, and X3 as independent variable, my understanding is that the predicted value is the value of Y at specified values of of X1,X2,X3. Ex. ...

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$
$$S_j=\frac{1}{n_j-1}\sum_{...

I have a data set which comprises N measurements. Each measurement is an 8 dimensional vector representing 8 voltages measured from a machine. I want to compute the covariance matrix of this data. ...

We know that the best 1st order approximation of an unconditional expectation is the following-
$$E(y|x)=(E(y)-\beta E(x))+\beta x$$
where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...

Suppose
$X$~$N(a,1)$
$Y|X$~$N(X,\sigma^2)$
Then what is
$X|Y<0$ ~ ?
and
$E[X|Y<0]$ ~ ?

I'd like some help setting up a probability model and determining it's EV, but I don't really know where to start.
Say there are 6 ranks, beginning at 0 and ending at 5.
For each rank ...

There is a question about EPE at StackExchange where next expresion is indicated:
$$\mathbb{E}[(Y-f(X))^2]=\mathbb{E}[(Y-\mathbb{E}[Y|X])^2]+\mathbb{E}[(\mathbb{E}[Y|X]-f(X))^2]$$
I don't ...

I am trying to characterize a sensor; in which I am trying to obtain the time constant as a function of the pressure applied to the sensor. I fitted a linear trend line into this data. However, I was ...

I am continuing my struggles with approximate Bayesian inference methods. I have a fundamental doubt about how to compute certain expectations that arise during variational bayes, for example.
So, my ...

I found an interesting coding challenge on pramp by a friend but I couldn't do it in time. Anyhow, it says given a set { 3,14,7,22,29,33} and random 3 element subset is generated each time and its ...

Suppose I have a distribution $\mathbb{F}$ with mean $M$. Also, assume we have a set of i.i.d samples of size $n$ denoted by $X=\{x_1, x_2,..., x_n\}$ from $\mathbb{F}$. As a result, all $x_1, ..., ...

This is a question from an old Ph.D Qualifying Exam.
Let $X,Y$ be two random variables. Find $g^{*}(X)$ so that $$\min_{g(x)}E(Y-g(X))^2=E(Y-g^{*}(X))^2$$
My attempt: If $g$ were a function of $Y$, ...

I'm trying to find the expectation of log(max/min) from n samples of Weibull(alpha, 1).
But I keep failing. Can anybody give some hints?
I tried:
$\mathbb{E}(\log(X_{max}/X_{min}) ) = \int_0^{\...

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 ...

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