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. For a discrete random variable, $X$, the expected value is $$ E(X) = \sum_{x} x P(X=x) $$ for a continuous ...
2
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2answers
53 views
How to compute expectation for poisson distributed variable in context of binomial?
I'm working on a problem that needs to be solved using EM algorithm. In doing that, I have to evaluate an expectation that I actually have no idea how to.
Consider: $Y$ as a a fixed observed integer ...
2
votes
1answer
53 views
Is Jensen's inequality applicable for two variables?
This is the Jensen's inequality I saw in my textbook:
$$E{ f(X) } \geq f( E(X) ),$$
where $f$ is a convex function.
Is this also applicable for two random variables--independent or otherwise--like ...
0
votes
0answers
22 views
help with proof involving conditional expectations
Attempting to understand a basic Lemma from this paper:
Lemma 8: Let $X, Y$ be i.i.d. random variables with values in an interval $[a,a+1]$, then
$$
\mathbb{E}_X\left[\mathbb{E}_Y (X - Y)^2\right]^2 ...
7
votes
1answer
136 views
Conditional expectation of R-squared
Consider the simple linear model:
$$\pmb{y}=X'\pmb{\beta}+\epsilon$$
where $\epsilon_i\sim\mathrm{i.i.d.}\;\mathcal{N}(0,\sigma^2)$ and
$X\in\mathbb{R}^{n\times p}$, $p\geq2$ and $X$ contains a ...
1
vote
1answer
38 views
Bivariate normal expectation of the sinus cardinal
I would like to get an analytical expression for
$$\mathbb{E}\left(\frac{\sin(aX)}{aX}\frac{\sin(bY)}{bY}\right)$$
or at least an analytical approximation thereof, when $a,b$ are positive reals, and
...
3
votes
0answers
43 views
Simple question about notation
In the context of likelihood-based inference, I've seen some notation concerning the parameter(s) of interest which I've found a little confusing.
For example, notation such as $p_{\theta}(x)$ and ...
1
vote
1answer
53 views
Expected value of latent utility in logistic regression
I'm looking for an analytical expression for the expected value of the latent utility in a logistic regression.
Setup:
There are two choices indexed by $i \in \{0, 1\}$ with associated utilities ...
3
votes
1answer
36 views
Expected number of duplicates (triplicates etc) when drawing with replacement
I have the following problem:
I have 100 unique items (n), and I'm selecting 43 (m) of them one at a time (with replacement).
I need to solve for the expected number of uniques (only selected once, ...
2
votes
3answers
87 views
Probability $X$ is less than mean
Let $X$ be a random variable with expected value $\mu$. I need an upper bound on $\Pr( X < \mu )$. All of the bounds I could find deal with something like $\Pr( X < \mu - a)$ and become ...
0
votes
0answers
18 views
Shrinkage estimator's risk function
How do you compute the risk function under squared loss of an estimator of the form
$\begin{align*}
\hat{\mu}(x) &= \bar{x} + \left(1-\frac{k}{||x-\bar{x}||_2^2}\right)(x - \bar{x})
\end{align*}$
...
0
votes
0answers
47 views
the expected value of your gain
You flip a coin and if it is a head I pay you 1 pound but if it is a tail you pay me 2 pounds. You have 50 pounds and you stop when you spend all of your money or you flipped coin 100 times. What is ...
3
votes
2answers
51 views
Confusion related to expectation
I was reading a paper related to minimaxity.
If $\hat{\theta}$ is the estimate of an unknown parameter $\theta$ and $l(\hat\theta-\theta)$ is the loss function they have mentioned $E_p$ is the mean ...
0
votes
2answers
151 views
Given a table defining the joint probabilities, how do I calculate certain parameters of the marginal distributions?
The number of items sold on any one day in the traditional shop is a random variable X and the corresponding number of items sold via the Internet is a random variable Y. The joint distribution of X ...
0
votes
0answers
88 views
Getting the MVUE of $\sigma^4$ for normal distribution
If x$_{1}$, x$_{2}$,...,x$_{n}$ ~ N($\theta$, $\sigma$$^{2}$), based on the fact that Y$_{1}$ = $\sum$(x$_{i}$) and Y$_{2}$ =$\sum$(x$_{i}$)$^{2}$ are complete and sufficient statistics, it can be ...
1
vote
0answers
68 views
Variance of powers of a random variable
Is it possible to derive a formula for variance of powers of a random variable in terms of expected value and variance of X?
$$\operatorname{var}(X^n)= \,?$$
and
$$E(X^n)=\,?$$
0
votes
0answers
113 views
Let X1,X2,…,Xn be i.i.d. N(θ1, θ2), please prove that E[(x1-θ1)^4] = 3θ2^2
If x$_{1}$, x$_{2}$,...,x$_{n}$ is sampled from N($\theta$$_{1}$, $\theta$$_{2}$), how can I prove that E [(x$_{1}$ - $\theta$$_{1}$)$^{4}$] = 3$\theta$$_{2}$$^{2}$?
I started off this question ...
6
votes
2answers
235 views
How to find the expected distance between two uniformly distributed points?
If I were to define the coordinates $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$ where
$$X_{1},X_{2} \sim \text{Unif}(0,30)\text{ and }Y_{1},Y_{2} \sim \text{Unif}(0,40).$$
How would I find the expected ...
3
votes
0answers
53 views
Testing for mutual exclusivity
So first, sorry if I use non-standard vocabulary, since I am not an expert. Please feel free to correct.
I would like to know if a set of genes in a patient cohort are mutated by a mutually exclusive ...
3
votes
0answers
65 views
In a Monte Carlo approximation of a product of expectations, can the same samples be used for both expectations?
I am trying to approximate a product of expectations:
$\mathrm{E}[f(x)]\mathrm{E}[g(x)]=\sum_x P(x) f(x) \sum_x P(x) g(x)$
with $N$ Monte Carlo samples $(x_1,x_2,...,x_N)$ from $P(x)$:
...
1
vote
1answer
101 views
How to find Coefficient of Variation with E(X), E(X^2) and sample size?
Given E(X), E(X^2) and sample size. How do I find Coefficient of Variation?
Variance = E(X^2) - E(X)^2
sd = square root of variance
Coefficient of Variation = sd / E(X)
however, with the values ...
2
votes
0answers
54 views
Finding correlation coefficient
if I have A and B with the following known variables:
with $E[A]$, $E[B]$ , $\sigma_{A}$ , $\sigma_B$
and correlation coefficient: $\rho_{AB}$ (assign numbers if you like)
Say: $C=0.6A+0.4B$
Then ...
0
votes
0answers
50 views
How is the formula for the expected value of a function of one random variable derived?
The "proof" in my textbook was very short and only in the discrete case, I didn't understand it very well too. So I tried to derive the expression starting from the expected value of a continuous ...
0
votes
1answer
96 views
How to generate an orthogonal vector
I have two vectors $a$, $b$ each containing $10$ random numbers from the standard normal distributions. I want to generate another vector $C$ of $10$ numbers from the standard distribution where ...
1
vote
2answers
187 views
Expected Value of Quadratic Form
On page 9 of Linear Regression Analysis 2nd Edition of Seber and Lee there is a proof for the expected value of a quadratic form that I don't understand.
Let $X = (X_i)$ be an $n \times 1 $ random ...
5
votes
1answer
215 views
What is the Fisher Information for the truncated poisson distribution?
The zero-truncated poisson distribution has probability mass function:
$P(X=k) = \frac{e^{-\lambda}\lambda^k}{(1-e^{-\lambda})k!}$, $k=1,2,...$
And the expectation of the truncated Poisson ...
6
votes
1answer
287 views
Central Moments of Symmetric Distributions
I am trying to show that the central moment of a symmetric distribution:
$${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf ...
3
votes
1answer
146 views
Expectation and confidence intervals of a Poisson process
A Poisson process has PDF
$$P(X=k)=\frac{e^{-\lambda t}(\lambda t)^k}{k!}$$
I'm trying to find an expression for:
$E[X | \lambda, t]$
Confidence intervals (i.e. find $\delta$ such that ...
-3
votes
1answer
743 views
expectation of an exponential function [closed]
What is the expectation of an exponential function:
$$\mathbb{E}[\exp(A x)] = \exp((1/2) A^2)\,?$$
I am struggling to find references that shows this, can anyone help me please?
I am assuming ...
1
vote
1answer
92 views
Asymptotic assumptions in OLS
Let us assume the following data generating process:
(1) $ \ \ y = X\beta + u$
where $y$ and $u$ are $n\times 1$ vectors, $X$ is a $n\times k$-matrix with $rk(X)=k$ and $\beta$ is a $k\times ...
1
vote
0answers
24 views
When are time averages equal to statistical averages?
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. ...
2
votes
0answers
116 views
Name for $E[X]^2/E[X^2]$?
Has anyone seen the following quantity come up in the literature?
$$\frac{\mathbb{E}[X]^2}{\mathbb{E}[X^2]}$$
I saw it in equation 10 of this paper.
4
votes
1answer
228 views
Expected value of a transformed random variable
I would be interested in computing the expected value of the following random variable $Y$.
Let $X$ be either a $\mathrm{Bin}(p,N)$ or a $\mathrm{Hyp}(n,m,N)$ with $m$ number of successes (I would ...
1
vote
1answer
237 views
How do you compute expected value of arbitrary distributions in scipy?
Suppose I have a python function using scipy that returns the expectation $E\left[ X \right]$ for some data assuming it is gamma distributed:
...
0
votes
0answers
88 views
How to express the expected value of revenue w/ reserve price in 2nd price auction?
Breifly speaking, the problem comes from setting a reserve price in 2nd price auction, where w/o a reserve price $a_t$, the winner pays the 2nd highest bid $b_t$. But w/ a reserve price, the winner ...
2
votes
0answers
47 views
Smooth expectations outside the exponential family
At page 85-86 of Young and Smith "Essentials of Statistical Inference" there
is an interesting result. If $X$ is a r.v. distributed according to the exponential family and $\phi(x)$ is a bounded (but ...
1
vote
0answers
99 views
MSE of filtered noisy signal - Derivation
I'm working on understanding the derivation of the optimal time constant for filters based on minimizing mean squared error. Unfortunately the text made a big jump between steps and lost me.
Here's ...
1
vote
1answer
54 views
Finding variance?
$\newcommand{\Var}{\mathrm{Var}}$
Consider $Z_i$ as a binary random variable with $\mathrm{Pr}[Z_i = 1] = \pi$. Also, consider $Y_i$ as:
$Y_i|Z_i = 0 \sim \mathrm{Poisson} (\lambda_0) $
$Y_i|Z_i = 1 ...
2
votes
0answers
29 views
The Effect of Confounders on OLS estimates?
I'd like to see in linear regression how not adjusting for confounders affects estimated coefficients?
Here is what my lecture notes say:
1) Un-adjusted model: $E[Yi] = \beta_0 + \beta_1X_i$
2) ...
4
votes
1answer
162 views
How to show operations on two random variables (each Bernoulli) are dependent but not correlated?
I was looking at the following question from "One Thousand Exercises in Probability" by Grimmett, page 25, question 16 (not homework just self-study):
Let $X$ and $Y$ be independent Bernoulli ...
0
votes
0answers
17 views
Expected value (Chi-Squared) [duplicate]
Possible Duplicate:
Why is sample standard deviation a biased estimator of $\sigma$?
Let $X_1, X_2, X_3 \sim N(0, d^2)$ and $T = X_1^2 + X_2^2 + X_3^2.$
T is chi-squared distributed, ...
2
votes
1answer
92 views
Conditional Expectation / Estimator Confusion
Let $X_1, X_2, X_3 \sim N(0, d^2)$ and $T = X_1^2 + X_2^2 + X_3^2.$
I have an estimator for $d$, $$\hat{d} = \frac{\sqrt{T\ 2\pi}}{4},$$
and another estimator for $d$, $$\tilde{d} = \frac{1}{3} ...
1
vote
2answers
85 views
Expectation of a minimum
Assume that $c_{ij}(W)$ is a simple function of a random variable W. (This comes is Dynamic Programming where we are currently at node i and we want to go to node j in the shortest path problem and W ...
1
vote
1answer
270 views
Conditional Expectation of Multivariate Distributions
My book states that $\mathrm{E}[X_2] = \mathrm{E}[\mathrm{E}[X_2\mid X_1]]$ and gives a weird proof and a short paragraph about how awesome this result is and how intuitive this is.
I neither ...
2
votes
0answers
67 views
Notation in GBM package vignette: expected value of loss functions
Can anyone help with the understanding of this notation (and idea) from the vignette for GBM in R?
It starts with the following:
Question 1: I believe this is simply saying that we are looking for ...
1
vote
1answer
268 views
Sum of Gamma distributions
The lifetime of a particular brand of batteries is known to have a gamma distribution. Tests on a large sample of these batteries show a mean lifetime of 480 hours and a standard deviation of 96 ...
2
votes
1answer
103 views
“Running it” multiple times in No-Limit Hold'em poker
Suppose on the flop in no limit hold'em two players go all-in. They both have 2 hole cards and there's 3 community cards on the board at present. There will be 2 more cards drawn and each player will ...
4
votes
2answers
157 views
When sampling without replacement from a given distribution, what's the total expected weight of the last k sampled items?
We are given a probability distribution $p_1 \ge$ $p_2 \ge$ $p_3 \ge \cdots p_n$ on $n$ items. We sample from this distribution without replacement (re-scaling probabilities after each sample to ...
-3
votes
1answer
66 views
Deriving expected value of the size of a set
I did an experiment with a die where I counted the number of occurrences of each value, and associated the values as an element in a set.
For example, here are the values I got when I rolled a die 18 ...
3
votes
2answers
349 views
Why does $\overline{y} = \hat \beta_{0} + \hat \beta_{1} \overline{x}$ in simple linear regression?
Today, once again, I observed that the dependent variable was predicted to be its mean when the independent variable was set to its mean in simple linear regression.
Let $(\hat{y},\hat{x})$ be ...
11
votes
4answers
532 views
Why is statistics useful when many things that matter are one shot things?
I don't know if it's just me, but I am very skeptical of statistics in general. I can understand it in dice games, poker games, etc. Very small, simple, mostly self-contained repeated games are ...
