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.

Filter by
Sorted by
Tagged with
2
votes
4answers
60 views

How do you check if expected cell counts are less than 5 when run Fisher's in R?

So I have gender (F,M) and blood pressure (Low, High) variables. I have 30, 100, 1, 5 in the observed cells. I understand that you use Fisher's exact test when one or more expected values are less ...
1
vote
1answer
33 views

What's the expected average time

We have the following situation: We are trying to find the path to a coffee shop. We have 3 streets when we exit our home and only one takes us to the coffee shop. If we take the 1st street, we make ...
2
votes
1answer
46 views

Expected value of $e^{vS}$, where $S$ is an exponential

I am studying queueing theory and in particular I am dealing with priority queues with preemption. I found this very interesting paper that treats various topics of interest. The system is composed ...
1
vote
1answer
76 views

$E[X|X=Y]=E[Y|X=Y]=E[X]=E[Y]$?

The last piece to comprehend this answer is to establish that $$E[X|X=Y]=E[Y|X=Y].$$ I assume $$E[X|X=Y]=E[X|\{\omega : X(\omega)=Y(\omega)\}]=E[X|A].$$ If we consider this the expectation of $X$...
0
votes
0answers
19 views

How to find the stock which will maximize profit in this specific question?

I have to tackle the following question and since I don't know the correct answer, I am not sure if I am heading in the right direction. The question is about a a newspaper seller who buys a ...
1
vote
1answer
20 views

Calculate a Conditional Expectation via Samples

Consider a binary random variable $Y \sim p(Y)$, a random variable $X \sim p(X)$ (can be discrete or continuous) and a conditional distribution $p(X|Y)$. Suppose that I generate $N$ samples from $p(Y)$...
0
votes
0answers
24 views

Expected value with respect to a conditional distribution

I am reading the thesis Variational Inference and Deep Learning. On page 16, while deriving the ELBO, the expectation is w.r.t. a conditional distribution $q_{\phi}(\mathbf{z} | \mathbf{x})$: $$ \...
1
vote
1answer
24 views

Finding the expectation: Draws to get 2 gold coins

Omar has saved 11 precious coins, 7 of which made of gold, in a jar. He draws the coins one by one from the jar, to find the gold coins. If $X$ is the number of coins drawn until he has found the ...
0
votes
0answers
24 views

Integral and expected value for multivariate distribution

for the last couple of days I have been struggling with a problem and I was hoping to get some help here. I have a function that looks as follows: $$ f(x,y,z)=\begin{cases} 0 & \text{if} \ (x*...
0
votes
1answer
28 views

Evaluating Word Embeddings: Expected Cosine Distance

One way to evaluate the quality of word embeddings is with tuples $(a, b, c, d)$ of words of analog relations of $a$ to $b$ and $c$ to $d$, such as ...
2
votes
1answer
35 views

Expectation of potential outcomes formula

In Mostly Harmless Econometrics, the author uses the following identity to derive an estimator for the causal effect: $$E \left[ \frac{Y_i D_i} {p(X_i)} \right] = E \left[Y_{1i} \right]$$ where: $...
0
votes
0answers
35 views

Formula for expected value of continuous random variable without using density

I'm looking for some correct notation. Consider the random variable $V$ with support $\mathcal{V}$ and probability distribution $P_V$. Consider a function $u:\mathcal{V}\rightarrow \mathbb{R}$. Let $...
0
votes
0answers
17 views

Is independence of two random variables sufficient to rearrange summations?

Suppose $X$ and $Y$ are two random variables where their sample spaces may or may not intersect. My question is whether there is ever a situation where the following is not true:$$\sum_{(X,Y)}p(x)p(y)(...
0
votes
1answer
23 views

Correlation coefficient of x and y

If we have $$ X\sim Poisson(\lambda), Y|X = x\sim Binomial(x+1,p) $$ What is the correlation coefficient of X and Y? So I used $$\rho=\frac{Cov(X,Y)}{\sqrt{Var(x)Var(Y)}} = \frac{E[X[E[Y|X]]-E[X]E[...
2
votes
2answers
57 views

Proof of (weak) consistency for an unbiased estimator

I want to prove a theorem stating: An unbiased estimator $\hat{\theta}$ of the unknown parameter $\theta$ is consistent if $V(\hat{\theta}_n$) $\to0$ for ${n\to\infty}$. I've tried using the ...
0
votes
1answer
38 views

How can we write the sample variance's formal definition of a continuous random variable considering Bessel's correction? [closed]

I am trying to find the formal way of writing the sample variance of a continuous random variable considering Bessel's correction. I ask because the sample variance is usually written this way: $$ ...
3
votes
4answers
45 views

How to make sense out of integration over discrete data points?

Looking for a proof of the expected value of the score function equating zero, I came to this document that was recommended in another answer. Considering that we have a sample of n x_i values, I ...
1
vote
1answer
23 views

when can integration and expectation be exchanged?

When is it possible to move expectations into integrals? In a proof of the Central Limit Theorem, at one point an expectation was moved into the integral (without much explanation of why that worked....
1
vote
1answer
36 views

A technical question about the reparametrisation trick

I was reading this post which enlightened me about the technicalities of the reparametrisation trick, but I only get the intuition of this equivalent transform and I'm not sure why it is true: $$𝐸_𝑞[...
0
votes
1answer
21 views

Practical Examples: Expectation of a function with respect to a probability

I have encountered the following phrasing while reading Bishop's "Pattern Recognition and Machine Learning": Although for some applications the posterior distribution over unobserved variables will ...
4
votes
1answer
70 views

Calculating the variance of a weight-average when the weights also have a variance

Assume there is a series of random variables $X_1$, $X_2$, ..., $X_N$ representing a series of values to be weight-averaged, and a corresponding series of random variables $W_1$, $W_2$, ..., $W_N$ ...
0
votes
0answers
4 views

Correct for known frequency of random sampling in prediction

When finding the expectation of a parameter $p$, depending on another parameter $h$, one likely wants to apply the formula $\hat{p}=E[p]=\sum_k E[p|h=k] P(h=k)$. Now, often the problem allows one to ...
1
vote
1answer
37 views

Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$?

Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$? In other words, if we have $E(X \mid Y,Z)=0$ then can we also say $E(X \mid Y)=0$?
2
votes
2answers
44 views

Double expectation (Not law of iterated expectation)

Suppose I want to calculate $$E_X[E_Y{g(X,Y)}]$$ Am I supposed to calculate $$ \int \int g(x,y) f(x) f(y) \,dx\,dy$$ or $$\int \int g(x,y) f(x,y) \,dx\,dy $$ where $f(x)$ is marginal pdf and $f(x,...
0
votes
0answers
29 views

Application of law of iterated expectations

I would like your help to show a statement that uses the law of iterated expectations. In my notation $Supp_X$ denotes the support of a random variable $X$. Consider the random variables $\epsilon,...
2
votes
1answer
46 views

Chi Squared Test to see if a treatment group spent more than a control group

I have two groups of people. One group were shown an ad and the other were not. I know in aggregate that the treatment group spent \$799 and the control group spent \$412, so overall it seems people ...
2
votes
0answers
15 views

Collective name for properties including expectation, variance, divergences, etc

Is there an established name for the class of properties of random variables such as expectation, variance, higher moments, divergences (e.g. divergence). These are properties of one or several ...
1
vote
1answer
61 views

Is the expected value of a RV same as the mean of the corresponding pdf?

As we know the expectation of a RV $X$ or a function, say $g(X)$, of $X$, both with pdf $p_{X}(x)$ is $$ \begin{array}{*{20}{c}} {X \sim {p_X}(x):}&{E[X] = \int {x.{p_X}(x)dx} }\\ {g(X) \sim {p_X}...
0
votes
0answers
7 views

Risk adjustments for outliers with extreme consequences

I have a vague memory of an adjustment to expected value calculations to account for highly improbable events with extreme outcomes. An example is the lottery. The odds of hitting it are ...
0
votes
2answers
34 views

Intuitive meaning of expected value

Wikipedia gives the intuitive meaning of the expected value as In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the ...
1
vote
1answer
27 views

Expected number of days?

You’re drawing from a random variable that is normally distributed $X \sim \text{N}(0,1)$, once per day. What is the expected number of days that it takes to draw a value that’s higher that two? ...
1
vote
1answer
37 views

Moment inequality: $E\mid X_1 X_2 X_3\mid \leq (E(\mid X_1\mid^3)+E(\mid X_2\mid^3)+E(\mid X_3\mid^3))/3 $ for zero-mean r.v.'s?

Let $X_1, X_2, X_3$ be zero mean random variables and assume $E(\mid X_i \mid ^{4+\delta})\leq C, i=1,2,3$ where $C$ is a constant and $\delta>0$ some positive small constant. How can I show that ...
0
votes
0answers
24 views

Can this be simplified $\mathbb{E}_{q(\vec{z} \mid \vec{x})}\left[ \log {p(\vec{x} \mid \vec{z})}\right]$?

Assume that $p$ and $q$ are two distributions and $x$ and $z$ are two random variables. Can the following term (which appears in the paper Auto-Encoding Variational Bayes) be further simplified? $$\...
1
vote
2answers
89 views

Is $\ln\{E[f(x)]\}$ equal to $E\{\ln[f(x)]\}$? [duplicate]

Is the logarithm of an expectation the same as the expectation of the logarithm?
0
votes
0answers
16 views

Conditional Expectation - confused as how to deal with Et[zt+1] - only know how to do Et-1[zt]

Consider the following MA(2) process: zt = ut + α1ut−1 + α2ut−2, where ut is a zero-mean white noise process with variance σ 2 (a) Calculate the conditional and unconditional means of zt, that is, Et[...
0
votes
0answers
31 views

Interpretation the law of total expectation

For a function of two RVs we have: $$ E[g(X,Y)] = E[\underbrace {E\{ g(X,Y)|Y\} }_{It\space is\space a \space function \space of\space Y}] = E[h(Y)] $$. For each $Y=y$, the inner expectation is the ...
1
vote
0answers
27 views

Literature on design of importance sampling distribution using MLE or point-estimates of highest modes

Suppose I have many distributions $p_i(\theta)$ I wish to take expectations over $$\mathbb{E}_{p_i}[\mathbf{f}_i(\theta)]$$ where the $\mathbf{f}_i$ are vector-valued. In my problem the $p_i$ share ...
0
votes
1answer
35 views

In spite of the condition $i \ne j$, why is $E[Y_iY_j] = \mu^2$?

I am following the derivation of the expected value of the sample variance https://en.wikipedia.org/wiki/Variance#Sample_variance In the penultimate step, they do the substitution $E[Y_iY_j] = \mu^2$ ...
1
vote
1answer
26 views

Unusual Markov inequality for normal distribution

I'm trying to answer the following question from Larry Wassermans book on statistical inference. My question is how did they arrive at the Markov bound, it does not seem like the normal form of the ...
1
vote
0answers
17 views

How to compute the expected number of events in the following conditional renewal process?

I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows: ...
0
votes
0answers
17 views

How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
0
votes
0answers
24 views

Expectation of a product of random variables problem

Let $U$ be a random variable with uniform distribution over $[0,1]$ and a bivariate random vector $(Z,T)$ defined by $$(Z,T)=(0,0)1_{\{U\geq 2\alpha\}}+(Q_X(U),Q_Y(U))1_{\{U< 2\alpha\}}$$ where $...
2
votes
1answer
136 views

How is this equation about expected value true?

$X$ is a stochastic variable. We have $E(X) = \mu$ and $Var(X) = \sigma^2$ Then $E(X^2) = \mu^2 + \sigma^2$ We have data $X_1, X_2, X_3 , \ldots, X_n$ from that distribution. Then we have: $$\sum_{...
0
votes
0answers
13 views

State value function $v(s)$ as an expectation [duplicate]

Suppose we have a Markov Decision Process with environment states $s \in S$, agent actions $a \in A$, and rewards $r \in \mathbb{R}$.So if an agent takes an action $a_t$ in state $s_t$, he will end up ...
0
votes
0answers
22 views

law of iterated expectations doubt

Is it correct to do so: $E(X^{-1} Y) = E_x(E(X^{-1} Y \vert X) = E_x(X^{-1} E(Y \vert X)) $ Are we allowed to use LIE in case of non linear functions like $X^{-1} $
1
vote
0answers
20 views

Deriving Value Function of a Markov Reward Process

I am looking through the coursework for a Reinforcement Learning course (I am not enrolled in it, this is for my own study). In the lecture notes, on page 6, equation 11, they provide the following ...
0
votes
0answers
30 views

expected value (Sheldon Ross 9th edition)

A person buy 10 lottery tickets with each having a winning probability p. If he wins a prize in atleast one of the tickets,he gets addicted and will keep on buying tickets till he wins again. Find the ...
0
votes
0answers
12 views

Expectation of a reciprocal of unbiased estimation [duplicate]

Consider the two "true" value of directions $\bf{n}$, $\bf{s}$ and their "unbiased" estimate ${\bf{n}}_0$ and ${\bf{s}}_0$. Let ${\bf{w}}={\bf{n}}^T{\bf{s}}/{\bf{n}}_0^T{\bf{s}}_0$, then the ...
0
votes
1answer
41 views

Showing the expectation of a lognormal AR(1) process

Suppose I have a lognormal AR(1) process: $$\log(y_{t+1}) = (1-\theta)c + \theta \log (y_t) + \varepsilon_{t+1},$$ $$\varepsilon \sim N(0,\sigma^2)$$ To show $\operatorname{E}(y_{t+1})$, is it ...
0
votes
0answers
43 views

What does E_n mean

I came across the following symbol in this paper: I am a little bit confused by the symbol . Does it simply mean population average?