Mathematics Stack Exchange
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Questions tagged [expected-value]

Questions about the expected value of a random variable.

The average value of a randomly chosen quantity is its expectation or expected value. For example, the expected value of the number you get when you roll a fair 6-sided dice is 3.5.

In general, if $X$ is a random variable defined on a probability $(\Omega, \Sigma, P)$, then the expected value of $X$, denoted by $E[X], \langle X \rangle,$ or $\bar{X}$ is defined as the Lebegue integral

$$E[X]= \int_{\Omega} X(\omega) dP(\omega)$$

The expected value is often the first and most important thing you want to know about a random variable. For example, in a betting game, the best strategy is often the one that maximizes the expected value of the amount you win.

This tag is for questions about:

  • Computing the expected value in a specific situation.
  • Understanding the properties of expected values, such as Markov's inequality or linearity of expectation.
  • Proving theorems about the expected value of abstract random variables.
  • Understanding what the expected value means and what it tells you about a random variable.
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Expectation $E[\frac{X_1 + X_2 + ... + X_k}{X_1 + X_2 + ... + X_n}]$, where $k < n$

Given a sequence $X_1, X_2, ..., X_n$ of i.i.d. random values, the problem is to find the expectation of $E[\frac{X_1 + X_2 + ... + X_k}{X_1 + X_2 + ... + X_n}]$, where $k < n$. There is no infromation about which distribution it is. We know that…
Nice guy
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Expectation of a random variable with exponents

If we let $A$ be equal to the number of tails we get in $n$ flips of a coin that is fair, what is the expected value of $A^2$. I am inclined to think that $E[A^2]$ is equal to $E[A]$ which is $ \frac{n}{2}$. Would this be the case?
Peter Peteson
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Does $E[X | Y] = 0$ mean $E[x | Y] = 0$ where $x$ is a realization of $X$?

I am given that $E[X | Y] = 0$ where $X, Y$ are both random variables. So this expectation is a random variable. By the tower rule we have $E[E[X | Y]] = E[X] = E[0] = 0$. Does this also imply that $E[x| Y] = 0$? Where $x$ is any realization of…
student010101
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Let X and Y have a joint uniform distribution on the triangle with corners at $(0, 2)$, $(2, 0)$, and the origin. Find $E(Y | X = 1/2)$

so $f(x,y) = 1/2$ on the region of the triangle. $f(x) = \int_0^{2-x}\cfrac{1}{2}dy = \cfrac{1}{2}(2-x)$ so $f(y|x) = \cfrac{1/2}{\cfrac{1}{2}(2-x)} = \cfrac{1}{2-x}$ So $E(Y|X=1/2) = \int_0^2y\cfrac{1}{3/2}dy = \int_0^2\cfrac{2}{3}ydy =…
user917163
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Bounds of integration for this expected value question.

Kelly throws a dart at a circular dartboard of radius $3$ feet. Let X and Y denote the location where the dart lands. Assume that $−3 ≤ X ≤ 3$ and $−3 ≤ Y ≤ 3$ and $X^2 + Y^2 ≤ 9$, i.e., the dart lands on the dartboard. Moreover, assume that the…
user908519
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Paying money to choose the box with \$100

There are 4 boxes. One box has $100. The other 3 has nothing. You pay \$X to choose any box. If it doesn't have the money, you can choose another box. How many should \$X be to ensure a fair game if the person plays optimally? The solution is…
roulette01
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Confusion about linearity of expectation

I am following the course CS 285 Deep Reinforcement Learning from UC Berkeley. In lecture 4, part 1 (around 17:00), prof. introduces an expectation of rewards of a trajectory over a policy. He first introduces the probability of a trajectory over…
iRestMyCaseYourHonor
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Expectation of a sum of equally distributed random variables

I'm studying the following term: $$W_{ij} = \frac{1}{n(n-1)E[f(X_1,X_2)]} \sum_{i\neq j}^n f(X_i,X_j)Y_j,$$ where $(X_i, Y_i)$ is a sequence of equally distributed random pairs and $f$ a measurable function. Question: Does…
Celine Harumi
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find an unbiased estimator of $\sigma$ based on $\sum X_i^2$

Let $X_1,...X_n$ be iid drawn from $N(\sigma,\sigma^2)$ where $\sigma\gt0$ is the unknown parameter. Find an unbiased estimator of $\sigma$ based on $\sum X_i$, find an unbiased estimator of $\sigma$ based on $\sum X_i^2$, and use these results to…
Vons
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Calculating expected value of a function

Consider the function : $f(L) = \sum\limits_{x:L} g(x)$ where L is a list of integers where each element is randomly chosen between $1$ and $20$ $g(x) = 0$ if $x < 10$ $g(x) = 1 $ if $10 <=x <=19$ $g(x) = 2$ if $x =20$ What is the expected value…
ishita
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Expected number of days

If it is a good day (G) there are 60% chances tomorrow will be G and 40% chances tomorrow will be bad (B). If it is a B day, there 30% chances tomorrow will be G and 70% chances tomorrow will be B. If today is B, what is the expected number of days…
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Using complementary CDF to compute partial expectation of a nonnegative random variable

I know that for a nonnegative random variable X, $E[X]=\int_{0}^\infty P(X>x) dx$. However, I want to calculate the expectation of $E[X\cdot\mathbb 1\{X>b\}]$ which is equal to $E[X]-E[X\cdot\mathbb 1\{X\leq b\}]$, with the complementary CDF of $X$.…
Nan Wang
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Expectation value of a geometric random variable. For dummies.

I am no mathematician, my syntax is crappy and I will know LESS than you think. Can anyone please help me with the following problem? Compute E(X) for the following random variable X: X=Number of tosses until getting 4 (including the last toss) by…
MDSv
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A Question about Expected Value

Let $p$ be a real number between $0$ and $1$. Simone has a coin that lands heads with probability $p$ and tails with probability $1-p$; she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she…
IncredibleSimon
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Expectation integral written in a possibly strange way

On page 4 of arXiv:1701.02434v2 ("A Conceptual Introduction to Hamiltonian Monte Carlo"), we have the following text and expectation: We begin by assuming that the target sample space, $Q$, can be parameterized by the real numbers such that every…
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