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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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Approximating expectation with Taylor series

I want to get the second-order Taylor approximation for an expectation. I have the following distribution, which is a Generalized Dirichlet distribution with parameters $\boldsymbol\alpha$ and $\...
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60 views

General solution of expected value of E(f(X))?

This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable ...
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43 views

Don't understand this Expected Value notation (E*)

I was shown this problem and I don't recognize the E* notation. Clearly (?) it's different from E for expectation, but related. Does anyone know this and/or point me to a suitable reference?
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How to check these sequences generated by i.i.d random variables are martingales?

Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables. $P(Y_i=1)=P(Y_i=-1)=\frac12$ Set $S_0=0$ and $S_n=Y_1+...+Y_n$ if $n\geq 1$ I want to check if the ...
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Expected value for max weight of two stones (given independent uncertainty in each) [duplicate]

If I have two stones A and B with estimated weights (and associated uncertainties) of A=100 +/- 5kg B=102 +/- 2kg Is there any formula (or good approximation) to compute the expected value of ...
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Expectation of kth order statistic of Pareto distribution

I am trying to find the expected value of $X_{(k)}$ Given cdf $$ F(x) = \begin{cases} 1-\left(\frac{\sigma}{x}\right)^\alpha, & x > \sigma\\ 0, & \text{else.} \end{cases}$$ My attempt: $$...
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18 views

Expectation of a function of random variables

I'm trying to simplify the following expectation so that I can later solve a maximization problem: $max_k E[(A - kB)^2]$, where $A$ ~ $N(0,\sigma^2_1)$, $B = A+ \epsilon$ and $\epsilon$ ~ $N(0,\sigma^...
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Expectation on estimator for Poisson distribution

I'm reading through the textbook "All of Statistics" and one of the problems gives the following estimator for the lambda parameter of the Poisson distribution: $\hat{\lambda} = \frac{\sum_{i=1}^n ...
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52 views

$E[\bar{X^3}]$ of N(μ,1)

Suppose $x_1, x_2, x_3,\ldots, x_n$ i.i.d. Normal(μ , 1) random variables with μ $\in$ $\mathbb{R}$ how can we calculate $E[\bar{X^3}]$
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99 views

Are i.i.d. random variables always associated with a distribution function? [closed]

It is known that independent and identically distributed (i.i.d.) random variables are mutually independent and each random variable has the same probability distribution as the others. However, is ...
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Proving $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the expected value of standard normal variable

I'm looking to prove that $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} z^2\, dz$ (where $Z$ is a standard ...
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Is there something analogous to probability matching for a single event? Allocating a bet over a dichotomous outcome

Probability matching refers to the phenomenon in which people who are making repeated guesses as to a particular outcome (e.g., rolling less than 5 on a 6-sided die) tend to match their number of ...
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11 views

Decide inventory to carry based on forecast model to max. profit

This is statistical modeling / forecasting optimization question. I have a forecast model which predicts units value for a year. Now if I carry more inventory than prediction, I lose all the extra ...
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1answer
109 views

Unbiased estimator of $\lambda(1 - e^\lambda)$ when $x_1,\ldots,x_n$ are i.i.d Poisson($\lambda$)

Suppose $x_1, x_2, x_3,\ldots, x_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. I was trying to find an unbiased estimator for $\lambda(1 - e^\lambda)$, but I could not ...
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40 views

Developing a simple scoring system for client performance

I was hoping someone can point me in the right direction to develop a simple scoring system or algorithm on clients revenue performance over a period of time. I have over 400 clients with their ...
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57 views

Expectation and Variance of dot product of a random vector and random linear combinations of vectors from the same distribution?

Let's say we have a multivariate distribution $D$ which generates random $n$-dimensional vectors $x$ for us ($x \in R^n$). We know that the dimensions of vector $x$ are correlated, and that each ...
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47 views

Expected value of a variable given a probability distribution

I have fitted some data with a normal and gamma distribution. But I need to find the expected value of my variable. For example is x is my data (mine is not normal) and fit them into a normal and ...
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36 views

Why am I expected to take n/m attempts to complete something with m/n odds?

I assume there are various ways of explaining it, but what's the most intuitive (yes I understand that is a relative concept) Edit due to request: Obvious example: There is a 1/6 chance of rolling a ...
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176 views

Double integral involving the normal CDF

I need to compute (or best approximate?) the following integral $$\int_0^\infty \int_0^\infty (1 + \alpha u)^{-1}(1 + v)^{-1} \Phi\left(\frac{\beta}{\sqrt{\gamma + uv}}\right) \text{d}u \text{d}v,\...
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58 views

Law of Iterated Expectations Example

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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Conditional expectation of uniform random variable given order statistics

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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89 views

Conditional expectation on exponential distribution

How can I compute the conditional expectation C-X in the following formula E[C-X|X$<$C]; where C is a constant and X a random variable following exponential distribution?
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Why is this a valid step (expectation w.r.t posterior)?

Reading through this paper and on page 10 they use the step: $$\int q(\theta|D,\phi) \log p(Y|X,\theta) d\theta = E_q \log p(Y|X,\theta)$$ Now obviously I understand why they have written this as an ...
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Bounds on Expectation $E[A(B-C)^2]$

[This question has been edited for more given conditions]. Given possibly correlated random variables $A,B,C$, I want to find the best upper bound for $E[A(B-C)^2]$ given the following: $E[A(B-C)]$ $...
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If I have less than 5 elements in a cell is there a way to use Chi Squared? [duplicate]

I have some data which has less than 5 elements in a cell, which would usually lead to fishers test being used instead. However, I'm unable to use fishers as there's not enough memory for it. chi ...
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1answer
33 views

Conditional expectation of two independent RV

The expectation of the product of two independent random variables $X$ and $Y$ is the product of the expectations: \begin{align} E(XY) = E(X)E(Y) \end{align} Let's add another random variable $Z$ in ...
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The expected occurences of successive draws

We throw a coin 1 000 000 times. How many times on average will make 13 successful heads? Now the problem with the naive: 1 000 000/(2^13) is that once it made 13 heads the 14 head will happen with 1/...
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25 views

Conditional Expectation Brownian Motion

So this is an exam question I had recently and I honestly had no idea on how to solve it. Let W(t) be a Brownian Motion stochastic process at time t with drift p and variance v^2 Let s exist such ...
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28 views

Bus arrival times and exponential distribution

I came across a question that is supposed to show us how the properties of the exponential distribution can be used. I know and have shown that $$P(X_i<min\{ X_1,\dots,X_n\})=\dfrac{\lambda_i}{\...
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1answer
37 views

Determine expected value for continuous random variable [duplicate]

I know that $E[X^n]$ is found by $$\displaystyle\int_{0}^\infty{x^nf_x(x)dx}$$ I simplified this to $$\displaystyle\int_{0}^\infty{ \frac{x^{\frac{v}{2}-1+n}e^{\frac{-x}{2}}}{\displaystyle\int_{0}^\...
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41 views

expected value of money for coin toss

I play the following game using a coin that lands heads with probability $p$. I start with $X_0$ = $1$ and at each stage I gamble all I have on the toss of the coin. If it lands heads I end up with ...
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2answers
39 views

Asymptotic Expectation of Ratio of Sample Averages

I have two random variables: $X$ and $Y$. I know that: \begin{equation} E[X]=E[Y]=\mu>0 \end{equation} I know that variance of both can be bounded: \begin{equation} \operatorname{Var}[X]<k, \...
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67 views

What is expected times of sampling that a full population is covered?

There is a population with N instances. A sampling means we draw randomly M instances from the population without replacement. ...
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Expected Classification error using indicator functions

I am given three variables (latter two are binary) $A,B$, and $C$, where $A$ = input vector, $B$ = whether data was chosen, and $C$ is the true label. $A$ and $B$ are conditionally independent given $...
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33 views

Expected value using indicator variable

Suppose that $8$ white balls and $2$ black balls will be randomly ordered, from left to right (with all permutations of the $10$ balls equally likely), what is the expected value of the number of ...
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33 views

Calculation of Variance — Difference between results by hand and R

When I calculate the variance by hand I get something different than Rstudio. They guy in the video however, did calculate it as I did, wrong. Why is that? My calculations: Observations $=1,2,3,4,...
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31 views

Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
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1answer
19 views

Expectation of multivariate gaussian w.r.t. other multivariate gaussian

I want to calculate the Kullbach Leibler Divergence of two multivariate Gaussians as in KL divergence between two multivariate Gaussians. At one point one has to solve the following expression (...
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1answer
42 views

Expectation of Sufficient Statistic

Consider $X \sim B(n,p)$ with pmf $P(X=x) = {{n}\choose{x}} p^x (1-p)^{n-x}$. The general exponential form of an exponential family distribution is $p(x|\theta) = f(x) g(\theta) e^{\phi(\theta)^T T(...
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26 views

Calculate number of points with expected values

The Problem Lets say you have a game where using an item gives you a certain number of points, with different probabilities for each point value. The probabilities are unknown, but can be estimated ...
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1answer
46 views

PMF and independence with two discrete random variables?

Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
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2answers
110 views

When is the optimizer of $\mathbb E[X]$ and $\mathbb E[X^2]$ the same?

Consider a non-negative random variable $X\sim p(\theta)$, that is, following distribution $p$ parametrized by $\theta$. Suppose we find a value of the parameters $\theta^*$ such that $$\mathbb E_{X\...
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45 views

Intuition behind gradient of expected value and logarithm of probabilities

I recently came across the following curious identity: $$\nabla_\theta \mathbb{E}_{x \sim D_\theta}[f(x)] = \mathbb{E}_{x \sim D_\theta} [ \nabla_\theta \log(D_\theta(x)) f(x)],$$ where $D_\theta$ ...
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1answer
9 views

An algorithmic fairness / equal calibration lemma

I'm having trouble with the lemma and proof on page 7 on this paper. Mitchell, S., Potash, E., & Barocas, S. (2018). Prediction-Based Decisions and Fairness: A Catalogue of Choices, Assumptions,...
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2answers
54 views

Expected value of $e^{sP}$ where s is a complex number and P is a Poisson rv

For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
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78 views

Conditional expectation of an exponential variable

I have trouble with conditional probabilities, therefore I am wondering if the following derivation is correct: Both X and Y are exponential random variables, with $\lambda_x$ and $\lambda_y$ ...
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1answer
55 views

What is the expected number of times you need to flip a coin before you see 2 heads? The heads do not need to be in a row

My attempt: The 2nd head must appear last in the sequence of k flips. Therefore the first head can appear in any of the first k-1 flips. The number of ways the first head appearing in the first k-1 ...
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1answer
51 views

Difference between empirical distribution and the data-generating distribution? [closed]

I understand that an empirical distribution is basically sampling from the sample set with replacement. However I am not quite sure how $ \hat{p}_{data} $ and $ p_{data}$ in Maximum likelihood ...
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38 views

How to calculate the expected value of a function of a Poisson variable

Let $Y \sim Poisson(\lambda)$, and $f(Y)$ is a function of $Y$. Is there a general method, either analytically or numerically, for calculating the expected value of $f(Y)$? In other words, I would ...
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17 views

Conditional probability given only the converse conditional probability, and the average of one variable

I’ve been working on this question for a few days now. Full disclosure: this is from a homework problem set. This is one of the exercises of Barnett's book on quantum information. A particle counter ...