# Questions tagged [expected-value]

The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.

1,852 questions
Filter by
Sorted by
Tagged with
0answers
34 views

### Continous version of an (almost) negative binomial distribution

Let a random variable $X$ be the number of independant Bernoulli trials needed to reach $s$ successes and $f$ failures when the probability of success is $p$. We therefore stop trials when we have $s$ ...
0answers
9 views

### Complete expectation of life in Demography

This is the question that I am trying to solve , I have done it halfway , but I am unable to comment on the nature of linear relationship between x and complete expectation of life
1answer
86 views

1answer
41 views

0answers
26 views

### Calculation of Variance from a 2 order Taylor expansion - Expecting a better estimation than with 1st order Taylor expansion

I tried to compute the variance of a squared ratio of 2 Gaussians random variables (not the same means and standard deviations between both). I generate the samples by Monte-Carlo method. I expect ...
0answers
27 views

### Why is this integral equal to $1$? (VBIL)

Let $p(y \mid \theta)$ be a likelihood and $\hat{p}_N(y \mid \theta)$ be an unbiased estimator of it. In VBIL they define $z = \log \hat{p}_N(y \mid \theta) - \log p(y\mid \theta)$ and call its ...
1answer
57 views

### Expected Value Contradiction

I am stuck with a simple expected value problem. Here it is: Assume I have 1 USD in savings now. I have to decide whether I keep my savings in USD or convert them into Euros. At the moment 1 USD = 1 ...
1answer
39 views

### Expectation of the log shifted-gamma

If $X \sim \Gamma(\alpha, \beta)$ and $c$ some constant then $Y=X+c$ follows a shifted Gamma distribution with pdf $$f_Y(y)=\frac{b^a}{\Gamma(a)}(y-c)^{a-1}e^{-(y-c)b}$$ $y\in[c, +\infty)$. What is ...
0answers
11 views

### Differentiating expected prediction error (EPE)

From Hastie-Tibshirani-Friedman p.18-19 $$EPE(f)=E(Y-f(x))^{2} = \int[y-f(x)]^{2}Pr(dx,dy)$$ If $f(x)\approx x^{T}\beta$ shows that by plugging in $f(x)$ in $EPE$ and differentiating w.r.t. $\beta$ ...
0answers
16 views

### Expected first passage time for random walk

A random walk on $\{0,1,2.....n\}$ with $p_{0,0} = p_{n,n} = 1$ and $p_{i,i+1} = p = 1-p_{i,i-1} = 1-q$ for $1 \leq i \leq n-1$ .Let $X_0 = i$ and $T$ be the first passage time to either 0 or $n$ ...
2answers
72 views

### Express expectation value of a joint distribution over a discrete and continuous random variable

Let $Y$ be a discrete random variable and let $X$ be an (absolutely) continuous random variable and $f(X, Y)$ a function of these two random variables. Let $P(X, Y)$ be the joint probability measure. ...
0answers
22 views

### Expected Value of a Fundraiser Raffle Ticket with Three Prizes

The raffle ticket above represents a local fundraiser. The first prize is a smart TV valued at 1200 dollars, the second prize is a cooler valued at 200 dollars, and the third prize is a 100 dollar ...
1answer
30 views

### Finding expected value from expectation of squared distance

This problem is actually a part of a much larger biology problem that I am working on. However, I will leave out the unrelated parts. Consider a sequence of points $\{(x_j, y_j)\}$ where neighboring ...
0answers
9 views

### Equivalence of Inverse Probability of Treatment Weights and Standardization: Hernan and Robins proof

Hernan and Robins provide a proof for the equivalency of inverse probability weights and standardization for estimating the potential outcome mean that I am struggling to follow (technical point 2.3, ...
7answers
2k views

### What distributions have an undefined mean but are not symmetric?

What distributions have an undefined mean but are not symmetric? I'm looking for a probability distribution function (and CDF) for which the mean is undefined, but not symmetric like Cauchy, but a ...
0answers
12 views

### Implications on the relation between signs of random variables

Consider a binary random variable $Z$ and a random variable $Y$. Suppose that the following relations hold $$Z=1 \Rightarrow Y\in \mathbb{R}^{+}_0\\ Z=0 \Rightarrow Y\in \mathbb{R}^{-}_0$$ In words, ...
1answer
38 views

### Calculating $\mathbb{E}^2(\sigma_t^2)$ where $\sigma$ is a GARCH(1,1) process

Given that $\alpha=0,113079$, $\beta = 0,873884$, $\omega = 0,0000081$ (and that $\text{kurtosis} = 235$), I need to calculate a call price using GARCH volatility: https://www.researchgate.net/...
1answer
35 views

0answers
30 views

### rank of an expected value of a matrix

x is (a * 1) vector y is (b * 1) vector x and y are independent then what is rank(E[xy']) I know that xy' should be (a*b) matrix and since they are independent. , however I am not sure about the rank
0answers
106 views

### The expected value of log Gamma function

Suppose $X$ is exponentially distributed with the rate parameter $\lambda$. If we have the expected value of $\log X$ as \begin{equation} \langle \log X\rangle=-\gamma-\log\lambda \end{equation} where ...
0answers
19 views

### Expected value of the sum of Bernoulli r.v. with p = cdf of Normal r.v

Let $y_1, y_2,\ldots, y_n$ be an independent samples from an unknown distribution and $q_p$ be its $p$th percentile. I constructed $B_i$'s such that $B_i \sim \text{Bernoulli}(F(y_i))$ where $F$ is ...
0answers
24 views

### The average of a random varible with pdf in the form of an parametric inegral

The pdf of a random variable $T$ in the interval $(0,1)$ in a certain problem I was trying to solve is given by : $$g(t)= c\int_{0}^{1-t} t^{m-1}\left[(u+t)^{m}-u^{m}\right]^{n-2}(u+t)^{m-1} d u$$ ...
0answers
67 views

### Information Matrix for Conditional Likelihood

I am studying the MLE theory on my own and I am confused by the difference between the fisher information matrix for the full sample and for one observation, when it comes to conditional likelihood. ...
0answers
24 views

### Self Study: Trivariate Normal Expectation with Inequality Condition

I'm reading a paper and found an interesting expectation. I know the result the author found but I can't figure out the intermediary steps because the author provided none. My attempt is getting ...
1answer
96 views

### Order Statistics: How to calculate expected value of a function involving first and second order statistics

I am currently stuck with a challenging problem. I have n values drawn i.i.d. from a distribution F(x). Let $v_1$ be the nth order statistic (highest value) and let $v_2$ be the n-1 order statistic (...
2answers
1k views

### Intuition of Random Walk having a constant mean

I am very new to time series analysis. A random walk is defined as $Y_t=\phi Y_{t-1}+\varepsilon_t$, where $\phi=1$ and $\varepsilon_t$ is white noise. It is said that process is non-stationary for ...
0answers
21 views

### Calculating expected value for a gacha game

I'm trying to do some sanity check on a gacha game I'm playing as I suspect their calculations are off. I've already done a monte-carlo simulation to verify my hypothesis, but I'm not sure how to do ...