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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54
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4answers
60k views

Why is expectation the same as the arithmetic mean?

Today I came across a new topic called the Mathematical Expectation. The book I am following says, expectation is the arithmetic mean of random variable coming from any probability distribution. But, ...
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Find expected value using CDF

I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing. ...
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4answers
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Taking the expectation of Taylor series (especially the remainder)

My question concerns trying to justify a widely-used method, namely taking the expected value of Taylor Series. Assume we have a random variable $X$ with positive mean $\mu$ and variance $\sigma^2$. ...
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10answers
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Deriving Bellman's Equation in Reinforcement Learning

I see the following equation in "In Reinforcement Learning. An Introduction", but don't quite follow the step I have highlighted in blue below. How exactly is this step derived?
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Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

This is an interview question for a quantitative analyst position, reported here. Suppose we are drawing from a uniform $[0,1]$ distribution and the draws are iid, what is the expected length of a ...
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6answers
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Why is the expected value named so?

I understand how we get 3.5 as the expected value for rolling a fair 6-sided die. But intuitively, I can expect each face with equal chance of 1/6. So shouldn't the expected value of rolling a die ...
33
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6answers
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Can somebody offer an example of a unimodal distribution which has a skewness of zero but which is not symmetrical?

In May 2010 Wikipedia user Mcorazao added a sentence to the skewness article that "A zero value indicates that the values are relatively evenly distributed on both sides of the mean, typically but not ...
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3answers
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Why not report the mean of a bootstrap distribution?

When one bootstraps a parameter to get the standard error we get a distribution of the parameter. Why don't we use the mean of that distribution as a result or estimate for the parameter we are trying ...
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3answers
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I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
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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 ...
26
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4answers
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Why maximum likelihood and not expected likelihood?

Why is it so common to obtain maximum likelihood estimates of parameters, but you virtually never hear about expected likelihood parameter estimates (i.e., based on the expected value rather than the ...
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5answers
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Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution?

It came as a bit of a shock to me the first time I did a normal distribution Monte Carlo simulation and discovered that the mean of $100$ standard deviations from $100$ samples, all having a sample ...
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How do I analytically calculate variance of a recursive random variable?

Suppose I have a chest. When you open the chest, there is a 60% chance of getting a prize and a 40% chance of getting 2 more chests. Let $X$ be the number of prizes you get. What is its variance? ...
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Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

For a uniformly distributed variable between 0 and 1 generated using rand(1,10000) this returns 10,000 random numbers between 0 and 1. If you take the mean, it ...
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5answers
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Let X,Y be 2 r.v. with infinite expectations, are there possibilities where min(X,Y) have finite expectation?

If it is impossible, what is the proof?
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Expectation of reciprocal of a variable

I am confused in applying expectation in denominator. $E(1/X)=\,?$ can it be $1/E(X)\,$?
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4answers
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Example of a non-negative discrete distribution where the mean (or another moment) does not exist?

I was doing some work in scipy and a conversation came up w/a member of the core scipy group whether a non-negative discrete random variable can have a undefined moment. I think he is correct but do ...
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Expected value of waiting time for the first of the two buses running every 10 and 15 minutes

I came across an interview question: There is a red train that is coming every 10 mins. There is a blue train coming every 15 mins. Both of them start from a random time so you don't have any ...
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3answers
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Expected number of tosses till first head comes up

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time. What is the expected number of tosses that will be required? What is the expected number of tails that will ...
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6answers
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Why is statistics useful when many things that matter are one shot things?

I don't know if it's just me, but I am very skeptical of statistics in general. I can understand it in dice games, poker games, etc. Very small, simple, mostly self-contained repeated games are fine....
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Expectation of a product of $n$ dependent random variables when $n\to\infty$

Let $X_1 \sim U[0,1]$ and $X_i \sim U[X_{i - 1}, 1]$, $i = 2, 3,...$. What is the expectation of $X_1 X_2 \cdots X_n$ as $n \rightarrow \infty$?
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1answer
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Conditional expectation of R-squared

Consider the simple linear model: $$\pmb{y}=X'\pmb{\beta}+\epsilon$$ where $\epsilon_i\sim\mathrm{i.i.d.}\;\mathcal{N}(0,\sigma^2)$ and $X\in\mathbb{R}^{n\times p}$, $p\geq2$ and $X$ contains a ...
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2answers
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Observed information matrix is a consistent estimator of the expected information matrix?

I am trying to prove that the observed information matrix evaluated at the weakly consistent maximum likelihood estimator (MLE), is a weakly consistent estimator of the expected information matrix. ...
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Why do we use the Greek letter μ (Mu) to denote population mean or expected value in probability and statistics

According to this Wikipedia entry, "Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water". So, my question ...
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What is the expected value of the logarithm of Gamma distribution?

If the expected value of $\mathsf{Gamma}(\alpha, \beta)$ is $\frac{\alpha}{\beta}$, what is the expected value of $\log(\mathsf{Gamma}(\alpha, \beta))$? Can it be calculated analytically? The ...
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Expected value vs. most probable value (mode)

The expected value of a distribution $f(x)$ is the mean, that is the weighted average value $$E[x]=\int_{-\infty}^{+\infty} x \, \, f(x) dx$$ The most likely value is the mode, that is the most ...
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Expected value of sample median given the sample mean

Let $Y$ denote the median and let $\bar{X}$ denote the mean, of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
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Why is the distribution of rand()^2 different than of rand()*rand()?

In Libre Office Calc, the rand() function is available, which chooses a random value between 0 and 1 from a uniform distribution. I'm a bit rusty on my probability, ...
15
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3answers
864 views

Why does the number of continuous uniform variables on (0,1) needed for their sum to exceed one have mean $e$?

Let us sum a stream of random variables, $X_i \overset{iid}\sim \mathcal{U}(0,1)$; let $Y$ be the number of terms we need for the total to exceed one, i.e. $Y$ is the smallest number such that $$X_1 +...
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2answers
867 views

Expected rolls to roll every number on a dice an odd number of times

Our family has recently learned how to play a simple game called 'Oh Dear'. Each player has six playing cards (Ace,2,3,4,5,6) turned face-up, and take turns to roll the dice. Whatever number the dice ...
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2answers
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Expectation of the Maximum of iid Gumbel Variables

I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then: $$E[\max_i(...
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1answer
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Notation: What does the tilde below of the expectation mean? [duplicate]

I am reading about variational auto encoders, and there is the below loss function: $$l_i(\Theta,\phi) = - {\mathbb{E}}_{z\sim q} \left[\log p_\phi(x_i|z)\right] + KL(q_{\phi}(z_i|x)||p(z))$$ What ...
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3answers
915 views

The frog problem with negative steps

Standard Problem description In this question The Frog Problem (puzzle in YouTube video) a frog has to jump from leaf to leaf on a row of leaves. And the question is how long it takes on average to ...
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1answer
1k views

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 ...
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5answers
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Probability that number of heads exceeds sum of die rolls

Let $X$ denote the sum of dots we see in $100$ die rolls, and let $Y$ denote the number of heads in $600$ coin flips. How can I compute $P(X > Y)?$ Intuitively, I don't think there's a nice way to ...
14
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2answers
92k views

How to calculate the expected value of a standard normal distribution?

I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is $$E[X] = \int_{-\infty}^{\infty} xf(x)\mathrm{d}x$$ where $f(x)$ is the ...
14
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1answer
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Expected value and variance of trace function

For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^...
14
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1answer
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What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ \left(\frac{X_1}{\...
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2answers
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Expected value of x in a normal distribution, GIVEN that it is below a certain value

Just wondering if it is possible to find the Expected value of x if it is normally distributed, given that is below a certain value (for example, below the mean value).
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1answer
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Why is ln[E(x)] > E[ln(x)]?

We're dealing with the lognormal distribution in a finance course and my textbook just states that this is true, which I find sort of frustrating as my maths background isn't very strong but I want ...
13
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1answer
513 views

OLS as approximation for non-linear function

Assume a non-linear regression model \begin{align} \mathbb E[y \lvert x] &= m(x,\theta) \\ y &= m(x,\theta) + \varepsilon, \end{align} with $\varepsilon := y - m(x,\theta)$...
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2answers
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Why is pseudo-random sampling applicable for Monte Carlo integration, even though it does not satisfy the CLT requirements?

Assume we have a function $f\left(x\right)$ defined on $\left[0, 1\right]$ that we want to integrate and estimate the error using Monte Carlo method. We generate realizations of uniformly distributed ...
12
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3answers
925 views

How do you calculate the expectation of $\left(\sum_{i=1}^n {X_i} \right)^2$?

If $X_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $X_i$'s are mutually independent, what is the expectation of $$ \left(\sum_{i=1}^n {X_i} \right)^2$$ in terms of $n$ ...
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2answers
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Constructing example showing $\mathbb{E}(X^{-1})=(\mathbb{E}(X))^{-1}$

How to construct an example of a probability distribution for which $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}$ holds, assuming $\mathbb{P}(X\ne0)=1$? The inequality which follows ...
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1answer
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Mean and variance of a zero-inflated Poisson distribution

Can anyone show how the expected value and variance of the zero inflated Poisson, with probability mass function $$ f(y) = \begin{cases} \pi+(1-\pi)e^{-\lambda}, & \text{if }y=0 \\ (1-\pi)\frac{\...
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2answers
1k views

Percentile Loss Functions

The solution to the problem: $$ \min_{m} \; E[|m-X|] $$ is well known to be the median of $X$, but what does the loss function look like for other percentiles? Ex: the 25th percentile of X is the ...
12
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2answers
735 views

Is the sample quantile unbiased for the true quantile?

I would like to find a way to show whether the sample quantile is an unbiased estimator of the true quantiles. Let $F$ be strictly increasing with density function $f$. I will define the $p$-th ...
12
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2answers
432 views

Expected value of spurious correlation

We draw $N$ samples, each of size $n$, independently from a Normal $(\mu,\sigma^2)$ distribution. From the $N$ samples we then choose the 2 samples which have the highest (absolute) Pearson ...
12
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1answer
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Expected value of $R^2$, the coefficient of determination, under the null hypothesis

I am curious about the statement made at the bottom of the first page in this text regarding the $R^2_\mathrm{adjusted}$ adjustment $$R^2_\mathrm{adjusted} =1-(1-R^2)\left({\frac{n-1}{n-m-1}}\right)....
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Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution?

If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread. Does this property have a name? Are there ...

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