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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Finding out expected number in penalty shootouts [on hold]

There are 3 expert players (Alex,Bob and Charlie) in a penalty shootout in a football team. The coach often has difficulty selecting an expert penalty shooter from the three expert players. Therefore, ...
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1answer
29 views

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 ...
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15 views

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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17 views

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 ...
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49 views

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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1answer
22 views

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 ...
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2answers
60 views

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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27 views

Regarding variance of “total number of records”

Here is my problem: Let $\{X_i\}_{i=1}^n$ be a sequence of continuous random variables. A record is said to have occured at time $k$ if $X_k > X_i$ $\forall i=1,...,k-1$. Let $N$ denote the ...
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22 views

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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16 views

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 ...
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24 views

Solving a problem related to expected value

After being all out for 58 and 78 in two matches in the most prestigious tournament in the world, the coach of a certain national cricket team was very upset. He decided to make the batsmen practice a ...
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20 views

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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1answer
35 views

KL divergence between a gamma distribution and a lognormal distribution?

Is there a closed-form formula for the KL divergence $D_{KL}(X,Y)$ where $X \sim \mathrm{Gamma}(k,\theta)$ and $Y \sim \mathrm{LogNormal}(\mu,\sigma^2)$ ? Many thanks.
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16 views

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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1answer
68 views

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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1answer
26 views

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 ...
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2answers
39 views

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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1answer
26 views

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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23 views

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 ...
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1answer
35 views

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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34 views

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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1answer
59 views

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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1answer
28 views

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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1answer
40 views

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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1answer
48 views

Expected survival time from log-logistic survival model in R from survreg

I'm currently estimating a survival model (accelerated failure time model) with a log-logistic distribution in R using the survival package and the survreg function. I want to simulate expected ...
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1answer
44 views

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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1answer
107 views

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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1answer
60 views

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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19 views

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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1answer
46 views

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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1answer
27 views

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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55 views

Can’t Find the Fisher Information of This Function

Can anyone help me find the Fisher information for this function: $$f(x|\lambda) = \lambda\,x^{\lambda-1}\quad \text{ where } \lambda \in [0,1]\,.$$ Thanks in advance!
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31 views

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 ...
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34 views

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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201 views

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 ...
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26 views

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 ...
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46 views

Finding Expectation of a Random Variable Using Its Joint Marginal Density

If X and Y have joint density function $$ f(x,y) = \frac{1}{y} \,\mathbb{I}_{0<x<y<1}, $$ how do I find the expectation of X or Y? Since E[X] requires us to know the PDF of X, I tried to ...
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40 views

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ . find $ E(U|W,X)$

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ find $ E(U|W,X)$ and conditions under which $E(U|W=w,X=x)$ is an increasing ...
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1answer
35 views

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. Thank you kindly.
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1answer
29 views

MSE decomposition to Variance an Bias Square

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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31 views

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 ...
2
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49 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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29 views

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 ...
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43 views

Definition of Expectation clarification

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 $A_1, \ldots, A_n$, ...
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14 views

Correlation co-efficient 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 outcome ...
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19 views

I don't understand the solution to this Chernoff inequality?

I have a sum, $S_n = \sum_{i=1}^n X_i$ of n iid Poisson distributed random variables $X_1,...X_n$ I am supposed to apply the Chernoff bound to $S_n$. My professor gave us the solution: However, I ...
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13 views

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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1answer
19 views

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 ...
4
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2answers
82 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 + ...