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.

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How to solve this Expectation of log of random variable

This may seem a trivial Question but I am confused and never come across this kind of expression where I need to simplify for a function of a random variable $R$. I have an expression $E\bigg [\frac{{...
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Learning Expected value

We probably have played the game "Throwing Balls into the Basket". It is a simple game. We have to throw a ball into a basket from a certain distance. One day we were playing the game. But it was ...
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Developing a heuristic for maximizing the "covering" of a distribution

Context There's a board-game called Settlers of Catan in which players compete to be the first to gain 10 victory points by trading various resources in exchange for pieces (or cards) worth victory ...
Geoff Little's user avatar
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Algebraic manipulation of $Var(Y|X)=E[(Y-E(Y|X))^2|X]$

Q: Show that $Var(Y|X)=E[(Y-E(Y|X))^2|X]$ is equal to $Var(Y|X)=E[Y^2|X]-(E[Y|X)]^2$. Answer: I know I have to use the law of iterated expectation to get to the second statement but I am having ...
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Penalty Shootout and Expected Value

There are 3 expert players(A,B and C) in a penalty shootout in a football team. The coach often has difficulty selecting an expert penalty shooter from the three expert players. Therefore, he makes a ...
Shahed al mamun's user avatar
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2 answers
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If $E[Y|X]=a$ for some constant $a\neq 0$, then does $cov(X,Y)=0$?

I'm currently working on the following problem: Q: If $E[Y|X]=a$ for some constant $a\neq 0$, then does $cov(X,Y)=0$? Now I am quite lost as to how to do this problem as the question does not ...
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Estimate E[x|A,B]: alternatives to bucketing for non-parametric estimation

I have a set of products. I would like to estimate Expected Value of items sold of the products wrt product price and age of the purchaser. One alternative is to assume a distribution and fit it. ...
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Finding a consistent estimator mathematically

This is my first post on this website so hopefully everything will go smoothly. Let me first ask the question, then go over my problem. Q: Suppose that we are given $({X_{1i},X_{2i}})$ which is a ...
John Hamm's user avatar
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Estimating counts from sampled data

I am working on counting events from sampled web logs. To formalize the problem, consider a random process in which we randomly record an event with known probability $r$. Say we have $n$ recorded ...
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KL divergence between a gamma distribution and a lognormal distribution?

Is there a closed-form formula for the following KL divergence? $D_{KL}(X,Y)$ where $X \sim \mathrm{Gamma}(k,\theta)$ and $Y \sim \mathrm{LogNormal}(\mu,\sigma^2)$
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Expected value of multiple events

There are three offers Offer A - $ 5 probability of redemption A - P (A) = 0.5 Offer B - $ 4 Probability of redemption B – P(B) = 0.6 Offer C - $ 3 probability of redemption C – P(C) = 0.7 If I ...
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Find the expectation and covariance of a stochastic process

The problem is: Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result ...
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Question regarding covariance

I'm trying to prove a theorem, where it is given that each $X_i$ is independent and identically distributed with mean $\mu$ and variance $\sigma^2$. Within this theorem, I have multiple sub-results to ...
Savage Henry's user avatar
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How to calculate $E[X^2]$ for a die roll?

Apparently: $$ E[X^2] = 1^2 \cdot \frac{1}{6} + 2^2 \cdot \frac{1}{6} + 3^2\cdot\frac{1}{6}+4^2\cdot\frac{1}{6}+5^2\cdot\frac{1}{6}+6^2\cdot\frac{1}{6} $$ where $X$ is the result of a die roll. How ...
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Deriving K-means algorithm as a limit of Expectation Maximization for Gaussian Mixtures

Christopher Bishop defines the expected value of the complete-data log likelihood function (i.e. assuming that we are given both the observable data X as well as the latent data Z) as follows: $$ \...
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Expected value of pair of success of

I am working on a problem from Harvard Stat 110 problem set. One of the problem (1.7.a) asks to find $$E\binom{X}{2}$$ , where random variable X is from hypergeometric distribution. What does the ...
Saurabh Saxena's user avatar
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Expectation of covariance in derivation of Kalman filter

I'm working through the derivation of the Kalman filter equations from this paper (or alternative source here) and I'm unsure of the derivation of the state prediction covariance (equation 2 in the ...
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AB test - Is it okay to use a result with a low confidence level

Suppose you conduct an A/B test of 10,000 views for each of version A and B, but the results take 3 months to capture. Despite a small number of views, achieving a goal (converting a "view" to a ...
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Probability - expected value and variance

"A man is playing versus a machine in the following way: The machine chooses 2 numbers randomly from the set of numbers 1,2,3,4,5, where a number can be chosen twice (with replacement). If the ...
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Maximum of uniformly distributed random variables using iterated expectations

I'm working through the problems in Wasserman's 'All of Statistics'. The chapter on expectations and conditional expectations ends with a (seemingly) easy problem: Let $Y$ be the maximum of $n$ iid ...
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Expected value of the inverse of a random variable

Let $X$ be a random variable. $X$ can take the value 1 with probability $p$, and the value 2 with probability $1-p$. Can we write $E[\frac{1}{X}] = \frac{1}{E[X]}$? (note that $E[X] \neq 0$) Thank ...
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Study design - fresh look!

Need to be advised outside of the circle. I am more a physiologist+mathematician plus-c,c++,java coder/developer. Chart data. From year 2001 till 2012. 89 nursing stations or emergency call ...
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Expected survival time from log-logistic survival model in R from survreg

I am currently estimating a survival model (specifically, accelerated failure time model) with a log-logistic distribution using the survreg function in the ...
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What is the expected number of coin flips, if you stop when the first coin flip is the same as the last?

In order to calculate the $\text{E}[X]$ where $X$ is the number of total coin flips, this is the approach I took: The probabilities are: $Pr(H) = p$ $Pr(T) = (1-p)$ Define indicator random ...
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Show that if $X \sim Bin(n, p)$, then $E|X - np| \le \sqrt{npq}.$

Currently stuck on this, I know I should probably use the mean deviation of the binomial distribution but I can't figure it out.
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Mean of predictive distribution

I observe independent, Poisson-distributed data $ D = \{x_1, ... x_n \} $ with mean parameter $ \mu $, i.e., $$x_i\stackrel{\text{iid}}{\sim}\mathcal{P}(\mu)$$ Over $ \mu $ I assume $ Gamma(\alpha_0, \...
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Find expectation or lower bound of log erf

I need to find the expectation of $\log \Phi(x)=\log \left(\int_{-\infty}^x\frac{1}{2\pi}\exp(-\frac{1}{2}s^2)ds\right)$. (I realise this isn't quite the error function, but not sure what to call it). ...
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Law of iterated expectations with two random variables

Let $X$ and $Y$ be two random variables. I want to calculate $E[X|X<Y]$. I am wondering whether I can use the law of iterated expectations in order to calculate it, i.e. $E[E[X|X<Y,Y]]$. Do I ...
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If $E[X(t)X(s)]=t \land s $.Show that this process has independent increments

Let $X(t), t\ge0$ be a real-valued Gaussian process with mean zero and covariance function $E[X(t)X(s)]=t \land s $.Show that this process has independent increments.
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Why is $E(u^2)=Var(y)$? (Binary Response Model)

I'm trying to show some results in binary response models, and a couple of the proofs use the "fact" that $E(u^2)=Var(y)$, but I can't see why this is. The setup is that $y$ takes on the value $0$ or ...
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Bounding the expectation of the difference between empirical vs generalization error

Let the (defect) difference between empirical and generalization error be: $$D[f_S] = I_S[f_S] - I[f_S]$$ where the empirical risk is: $$I_S[f_S] = \frac{1}{n}\sum^n_{i=1} V(f_S,z_i)$$ where $V(f,...
Charlie Parker's user avatar
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Calculate expected value from discrete beta distribution

I have a lookup table in 2 variables, $Z_l$ and $T_l$. So, $Z_l$ and $T_l$ are vectors with same length where $Z_l$ goes from 0 to 1 and $T_l$ varies between 300 and 2000. If you are curious, $Z_l$ ...
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Expected value of a product of two compound Poisson processes

I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone ...
M Tegling's user avatar
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Mean square convergence

I am working on an example in my book and cannot figure out an expectation. Let $$E(T_n)= \frac{n\theta}{n+1}$$ $$E(T_n^2)= \frac{n\theta^2}{n+2}$$ $$g(t)=\frac{nt^{n-1}}{\theta^{-n}}$$ Then $$E((...
Joel Sinofsky's user avatar
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1 answer
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Expected value of least squares estimator $\hat{\beta}$

Given $\hat{\beta} = (X^{T}X)^{-1}(X^{T}Y)$, how do you derive the expected value? I found answers for finding the variance matrix but not the expected value.
Kyle's user avatar
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42 votes
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MSE decomposition to Variance and Bias Squared

In showing that MSE can be decomposed into variance plus the square of Bias, the proof in Wikipedia has a step, highlighted in the picture. How does this work? How is the expectation pushed in to the ...
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Bound on the expectancy of the maximum level in skip list

Let $M$ be a random variable for the maximum level of skip list, $M$ is a positive integer, $k$ is an integer from 0 to $\infty$, and $$ \Pr(M>k) = 1 - (1-p^k)^n \leq np^k $$ In the article Skip ...
TheLastOfTheMoops's user avatar
2 votes
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391 views

Probability that uniformly distributed points in a square region form a cluster

I have a known number of points N uniformly distributed in a square and I want to solve the expected number of clusters of points. I cluster is formed by a growing algorithm. Starting at a point p, ...
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Improper use of an expectation?

A derivation in a paper (theoretical ecology--there are often mathematical errors there) I am reading essentially uses the following line: $\frac{1}{n}\sum_{i=1}^{n}X_{i}=E\left[X_{i}\right]$. This ...
Biomath's user avatar
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Definition of Expectation clarification [closed]

In Econometric Theory of Davidson (2004) I read (p. 446): In terms of the parent probability space $(\Omega, \mathcal{F}, P)$ this implies a partition of $\Omega$ into sets $A_1, \ldots, A_n$, ...
Konstantinos's user avatar
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Correlation coefficient calculation

Suppose an experiment having $r$ possible outcomes $1,2,\dots,r$ that occur with probabilities $p_1,p_2,\dots,p_r$ is repeated $n$ times independently. Let $X$ be the number of times the first ...
shadow10's user avatar
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Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
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1 answer
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Expected value of a dice game

Say that I have a dice game. You can roll the die first and then have two choices. First, take the dollar amount of the number that shows up (if you rolled a 5, you get $5 ). Second, you can re-...
Paul Smith's user avatar
7 votes
2 answers
16k views

Expectation of $(X + Y)^2$ where $X$ and $Y$ are independent Poisson random variables

I would really appreciate anyone's help with this problem: (let $E$ denote expectation) Suppose $X$ and $Y$ are independent Poisson random variables, each with mean $1$. Find: $E[(X + Y)^2]$ ...
Lindsey's user avatar
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3 votes
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You will randomly select 10 balls from the box with replacement what is $E(\bar X)$

A box contains 100 numbered balls - 21 with the number 1, 36 with the number 2 and 43 with the number 3. You will randomly select 10 balls from the box with replacement and you take the mean of the ...
Bob Unger's user avatar
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1 vote
1 answer
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Expected lifetime of a device with two parts each having spares?

Consider a device with two parts : (1) and (2). Part (1) has 2 spares and part (2) has one spare. Lifetime of part (1) and its spares have iid exponential distribution with rate lambda. Lifetime of ...
Bob's user avatar
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16 votes
1 answer
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Expected numbers of distinct colors when drawing without replacement

Consider an urn containing $N$ balls of $P$ different colors, with $p_i$ being the proportion of balls of color $i$ among the $N$ balls ($\sum_i p_i = 1$). I draw $n \leq N$ balls from the urn without ...
a3nm's user avatar
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4 votes
1 answer
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Can someone provide an proof for $E[P[A|X]] = P[A]$

I'm tired of seeing the word "trivial" for this equality on every single lecture notes I could find online. Can someone please show me why this is indeed trivial? Thank you!
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2 votes
1 answer
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Variance of product of two random variables

I’m trying to calculate the variance of a function of two discrete independent functions. The first function is $f(x)$ which has the property that: $$\Bbb{P}(f(x)) =\begin{cases} 0.243 & \text{...
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3 votes
1 answer
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Calculating the expected value and variance of an estimator of a normal quantile

I don't quite understand how to use the estimator function and the variance function and plug in the sample mean. I expected that we would plug in the value $\bar X - 1.645s$ into $E(s)$ and $V(s)$. ...
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