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
0 votes
0 answers
4 views

Expected Value and Variances [closed]

"Suppose that Y1, Y2, Y3, and Y4 are four independent random variables, with E[Yi]=1 and Var[Yi]=2. Using the rules for expected value and variance, calculate the expected value and variance of ...
  • 1
0 votes
0 answers
11 views

How do I calculate the expected values if the correlation coefficients are -1/0/+1? [closed]

How do I calculate the expected values if the correlation coefficients are -1/0/+1? I know that -1 means backwards correlation and so on but I just need an example.
  • 1
0 votes
0 answers
24 views

Law of Iterated Expectation for a Probability

In my understanding of a derivation, the following statement seems to be used: $$Pr[V\leq X]=E[Pr[V\leq X|X]]$$ where both $X$ and $V$ are random variables. Is this a kind of the law of iterated ...
  • 711
1 vote
1 answer
52 views

Expected value of x to the power of a [closed]

I would like to understand if $\mathbb{E}(x^a)$ is equal to $\mathbb{E}(x)^a$ if $0<a<1$. Generally I know that $\mathbb{E}(x^a) \neq \mathbb{E}(x)^a $ but under which conditions this could be ...
0 votes
0 answers
14 views

Expected Num of Good Pairs in a Fully Connected Graph

I got the following question in my interview: there is a fully connected graph with each edge has a weight $w$, i.i.d generated from an unknown distribution $F$. A pair $(a, b)$ is called a good pair ...
  • 203
3 votes
0 answers
40 views

What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$

Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
0 votes
0 answers
16 views

Calculating expected value of $E[x(t)x(t+\tau)]$

Assuming $x(t)$ is a time invariant function. How do I calculate the expected value of $E[x(t)\,x(t+\tau)]$? Is calcuating it as follow correct? \begin{align} E[x(t)\,x(t+\tau)] &= E[x(t)] \cdot E[...
10 votes
5 answers
1k views

Expectation of random sum of non-random numbers

I have a continuous random variable $\tau$ and I want to evaluate $$ E\left(\sum_{i=1}^{\lfloor \tau \rfloor} Y_i\right), $$ where $Y_i$ are known, non-random, and $\lfloor . \rfloor$ is the floor ...
8 votes
1 answer
199 views
+50

Does $E\frac{1}{\|x\|^{4}} \rightarrow \frac{1}{E\|x\|^4}$ in high dimensions?

Suppose we have a Gaussian distribution centered at zero, covariance matrix $\Sigma$ with $\operatorname{Tr}\Sigma=1$ and $\operatorname{Tr}\Sigma^2=\frac{1}{2}$ When I try various sequences of such ...
0 votes
0 answers
19 views

Mean of $t$ in the logistic regression model

The question comes from a sentence of page 208 of Christopher M. Bishop's "Pattern Recognition and Machine Learning". The sentence is excerpted as follows: My questions are mainly two: Why ...
  • 55
1 vote
0 answers
37 views

Compute Expectation of Product Using Joint Survival Function

I know that, for a nonnegative random variable $X$, $$ E[X] = \int x dF(x) = \int S(x) dx$$ where $F(x)$ and $S(x)$ are the CDF and survival function of $X$, respectively. This was derived using ...
  • 215
0 votes
0 answers
16 views

Expectation Value of The Product of Two Variables [duplicate]

If I wanted to take the expectation value of Z where X and Y are independent say $Z$ $=$ $XY$ - $\mu_x$$Y$ and $\mu_x$ $=$ $E[x]$ also $\mu_y$ $=$ $E[y]$ I'd first write $E[Z]$ = $E$$[$$XY$ - $\mu_x$$...
1 vote
0 answers
14 views

Is this a property of the expected value and variance? [closed]

If Z=X+Y and X and Y are independent and X~N($\mu_X$,$\sigma^2_X$) Y~N($\mu_Y$,$\sigma^2_Y$) Can I say that E[Z] = E[X] + E[Y] and Var[Z] = Var[X] + Var[Y] I think this may be an insanely simple "...
0 votes
1 answer
41 views

Probability that next flip is tails, given $P(p ≤ z) = z^4$ + last five flips were tails

Bob gives Alice a weighted coin that lands on tails with some probability $p$. Specifically $P(p \le z) = z^4$. Alice flips the coin 5 times, and every time it lands tails. Find the probability that ...
2 votes
1 answer
61 views

Difference between measuring rates as $E(X/Y)$ vs. $E(X)/E(Y)$?

I'm working on a difference-in-differences (DiD) analysis for a healthcare study measuring # hospitalizations (X) per # eligible months (Y). Simplifying the math a bit - my actuary colleagues like to ...
  • 4,686
0 votes
0 answers
28 views

Square of a zero random variable

Following up on this question and the answer: Bootstrap variance of squared sample mean Summary: $X_1$, ..., $X_n$ are IID. Define $$ \overline{X}_n=\frac{1}{n}\sum X_i $$ and $T_n=\overline{X}_n^2$. ...
  • 131
1 vote
0 answers
27 views

Likelihood function-expectation

Given likelihood is a function of parameters, I cannot understand why the expectation of likelihood functions is not calculated with respect to the the parameter space but the sample space, as put ...
  • 11
30 votes
10 answers
4k views

Sample two numbers from 1 to 10; maximize the expected product

Assume you sample two numbers, randomly drawn from 1 to 10; you could choose two strategies: 1) pick with replacement and 2) pick without replacement. Which strategy would you prefer to maximize the ...
2 votes
1 answer
53 views

Expectation given pairwise covariances

I have 4 variables A,B,C,D over {-1,1} (Rademacher variables) and know that ...
  • 21
2 votes
0 answers
31 views

Why does $P_0^{(n)}(\phi_n) \leq e^{-Cn}$ implies that $\sum_{n\geq 1} P_0^{(n)}(\phi_n > e^{-Cn}) < \infty$?

In the proof of Schwartz's Theorem, the author makes the following statement By Markov’s inequality, the assumption $P_0^{(n)}(\phi_n) \leq e^{-Cn}$ implies that $\sum_{n\geq 1} P_0^{(n)}(\phi_n > ...
  • 83
5 votes
1 answer
77 views

Equality in Gaussian Poincare Inequality

The Gaussian Poincare inequality states that: for $f: \mathbb{R}^n \to \mathbb{R}$ and $Z\sim \mathcal{N}(0,I)$, we have that \begin{align} Var(f(Z)) \le E[ \| \nabla f(Z)\|^2]. \end{align} My ...
  • 203
21 votes
2 answers
2k views

When mathematical statistics outsmarts probability theory

This is not a question, but it is too good to pass. I read it is originally due to Enis, Peter. "On the relation $E (X) = E [E (X∣ Y)]$." Biometrika 60, no. 2 (1973): 432-433. Assume $Y$ has ...
0 votes
1 answer
37 views

Is it valid to move a logarithm inside of an expectation?

I have the following derivation for a latent variable model. $$ \newcommand{\d}[1]{\mathrm{d}#1} $$ $$ \begin{align*} \mathbb{E}_{q(x_0)}\left[-\log p_\theta(x_0)\right]\\ &=^{1.} \mathbb{E}_{q(...
  • 227
1 vote
1 answer
34 views

When is a probability a function of a probability?

I am asked to calculate $E(ln (\frac{1}{P(X)}))$ Where X is a discrete random variable over $\chi $ $\in \{1,2,3\}$ with probability mass function P(X=1)=0.5,P(X=2)=0.4, P(X=3) = 0.1 I think this ...
  • 479
-1 votes
1 answer
47 views

$cov(X,f(X))\neq 0$ and $E(X f(X))\neq 0$

Take a random variable $X$. Is it true that (1) $cov(X,f(X))\neq 0$ for any function $f$? (2) $E(X f(X))\neq 0$ for any function $f$? I believe the answer to both questions is no. However, can you ...
  • 421
0 votes
0 answers
17 views

An Equivalence Statement of Two Expected Values

Suppose that we have three random variables, $V_1,\;V_2,\;V_3$. I derived an equivalence statement through a pretty long derivation: $$\mathbb{E}[V_1|V_2=v_2,\;c>V_3]=\mathbb{E}\left[\;\mathbb{E}\{...
  • 711
0 votes
0 answers
10 views

What is the relationship between the manifest correlation between ranked variables and the latent, continuous correlation?

Suppose you draw n pairs of observations from a real-valued bivariate distribution. You then convert each observation to its ascending rank order. Given the population correlation for the real-valued ...
  • 122
3 votes
1 answer
70 views

Why is the expectation of a random vector still a vector?

In my previous post on derivation of covariance between y and random effect, for the following linear model: Frank's anwer proved cov(y, u) = ZD as below: Frank's proof involved E[u]. As far as I ...
  • 197
3 votes
4 answers
593 views

Why do we need importance sampling?

Let's say we want to calculate the following expectation: $$ \mathbb{E}_{z\sim p_z(z)}[f(z)] $$ One issue, is that the samples from $p_z(z)$ could be not very informative: We see here that $f(z)$ ...
  • 227
3 votes
2 answers
50 views

What is the expected value of this process?

There are $n$ piles each containing $a_i$ stones. In a sequence of moves, Alex chooses two neighbouring piles randomly (containing, say, $A$ and $B$ stones) and combines them to create a single pile ...
0 votes
0 answers
16 views

How to calculate an expected value of this expression?

I have three random variables: $X$, $Y$, and $Z$ and I know their expected values. I want to calculate: $$E\left[\frac{3-X-Y}{Z}\right].$$ I know that $E\left[\frac{1}{X}\right] \neq \frac{1}{E\left[ ...
9 votes
2 answers
709 views

How to simulate the St. Petersburg paradox

I am trying to find the expected value of the game in the St. Petersburg paradox. The game has infinite expected value, but in my simulation the payouts are about 15 on average, which is way too small....
  • 93
0 votes
1 answer
35 views

Expectation of the log of a negative random variable

I read the following result based on a second order Taylor expansion (here: Expected value of a natural logarithm) $$\mathbb{E}(\log(x)) = \log(\mathbb{E}(x))-\frac{1}{2}\frac{\mathbb{V}\text{ar}(x)}{\...
0 votes
0 answers
25 views

Expected value of max of $k$ linear combinations of Bernoulli random variables

Let $X\in\mathbb{R}^{n\times k}$ be a matrix with entries $x_{ij}$ and $(B_1,\dots,B_n)$ denote $n$ independent Bernoulli random variables with probability of success $p$, and furthermore let $Y=\max_{...
  • 23
0 votes
0 answers
46 views

Law of large numbers for transformed and non-transformed random variables

Law of large numbers states that: If $X_1, \dots X_n \sim p(x)$ are IID, then $ \frac{1}{n} \sum_{i=1}^{n} X_i \rightarrow \mathbb{E}_{p(x)}\{X\}= \mu$, where $X \sim p(x)$. Below is what I'm having ...
0 votes
0 answers
12 views

sum of expected differences in time series

I have a (markov) decision process without reward. I am estimating the expected state differences $\mathbb{E}[\delta_t]$ where $\delta_t = X_{t+1} - X_t$. A state $X_{t+1}$ can be expressed as $X_0 + \...
0 votes
0 answers
28 views

What is the sum $\sum_{m} e^{i (U_m k + \beta_m)} $ when $U$ and $\beta$ follow different distributions

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ $i = \sqrt{-1}$ Here, $U_m$ are samples drawn from a Gaussian random distribution. $$ U_m \sim \mathcal{N}(\mu, \sigma) $$ ...
1 vote
1 answer
36 views

Positive expected value for lottery

As far as I know, to decide either you should enter a bet or not, you should get the expected value of that bet I was wondering if the lottery has a very high expected value, is it wise to join? There ...
's user avatar
0 votes
0 answers
52 views

Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable

Consider a random variable \begin{align} U= a \log (Z+b)-Z \end{align} where $a,b>0$ and $Z$ is an exponential random variable. Question: Can we find the variance of $U$? Things that I tried ...
  • 203
2 votes
1 answer
33 views

What is the expectation of the exponential distribution multiplied by indicator function?

I am reading the research paper [A New Bayesian Lasso], where $u$ has the distribution The expectation of $u_j$ is given by $$\frac{1}{\lambda}+|\beta_j|$$. I know that the term $\frac{1}{\lambda}$ ...
0 votes
2 answers
163 views

Expectation of a Hebbian term

In a paper by Akrout et al., it is mentioned that [Given $\mathbf{y} = \mathbf{W}\mathbf{x}$, where $\mathbf{x}\in\mathbb{R}^m$ is an input vector, $\mathbf{W}\in\mathbb{R}^{m\times n}$ is a matrix, ...
  • 595
0 votes
0 answers
19 views

Expected value of a product of two ReLU functions

Let assume $\theta,\theta'\in R^d$ (are two fixed $d$ dimensional real-valued vectors) and $x\sim N(0,I_d)$ (is a d-dimensional random vector comes from standard normal distribution). Now, I am ...
  • 31
2 votes
2 answers
74 views

Is the expected value of a probability over an interval meaningful?

I am reading an unpublished manuscript and have come across an equation of the following form for the calculation of the probability of an even A, $$ P[A]=E\Big[P[X>x|Y]\Big]. \tag{1} \label{1} $$ ...
  • 123
19 votes
9 answers
3k views

100-sided dice roll problem

When should I stop rolling if it costs $1 for each roll and I earn only the value of the final roll shown on a 100-sided dice roll? My intent is to maximise profit and I have unlimited rolls
  • 191
0 votes
0 answers
32 views

Expected value of a complicated function

I want a closed form/ semi closed form of the expected value of a complicated function. The function looks like this, $$ f(x) = \frac{\sin(A \frac{x}{2})}{\sin(\frac{x}{2})} \frac{\sin(M B\frac{x}{2})}...
3 votes
2 answers
152 views

Implication of $E(XY)\neq 0$

Consider two real-valued scalar random variables $X,Y$ such that $$ (1) \quad E(XY)\neq 0,\quad E(X)= 0, \quad E(Y)= 0 $$ Let $g: \mathbb{R}\rightarrow \mathbb{R}$ be a function. Under which ...
  • 421
0 votes
0 answers
12 views

Expected number of random cell visits in Q-learning

I'm interested in Q-learning with a time-decaying exploration rate $$ \varepsilon_t = \exp(-\beta t)$$ In order to evaluate how `reasonable' certain values of $\beta$ are, and to make them comparable ...
  • 1,025
0 votes
0 answers
21 views

Expected value of [variance of sample means] is the average of [sample mean variance]?

This is drawn from Gelman et al in Regression and Other Stories: If I take $k$ independent sample proportions $p_i$ with differing sample sizes $n_i$, Gelman et al says that the observed variance of ...
  • 1
7 votes
2 answers
82 views

Expectation of truncated distribution

Consider the random variables $X,Y$ and assume that $$ E(X|Y)=0 $$ Does this imply that $$E(X|X\geq A,Y)\neq 0 ?$$ I think this holds for the truncated Normal, for example. But does it hold ...
  • 421
1 vote
1 answer
25 views

In an RCT, does running OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ and recovering $\tau$ recover ATE or ATT

Let's say I run an RCT and then run OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ where $D_i$ is a dummy variable indicating whether an individual $i$ received the treatment. If I were to take the ...
  • 11

1
2 3 4 5
42