Mathematics Stack Exchange
Mathematics Stack Exchange

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.
6766 questions
0
votes
1 answer

Calculation of expected value.

Let N be a random variable with the following distribution: $$ P(N=n)=(n+1)\left(\frac{3}{4}\right)^{2}\left(\frac{1}{4}\right)^{n}, n=0,1,2,...$$ Let $\xi_{1},\xi_{2},...$ be a sequence of independent random variables with the same Bernoulli…
A.Fue
  • 11
0
votes
0 answers

Expectation to draw all numbers from 1 to 100

I have 100 papers each contains numbers like (1,2), (2,3),(3,4),..., (99,100), (100,100). I draw each paper and note down its numbers and return that paper. How any times I have to draw on average to note down all numbers from 1,2, .., 100? …
str
  • 109
0
votes
3 answers

Find the maximum value of $x+(p/x)$, if $x<0$ and $p>0$

I tried to solve it this way: $$x+(p/x)=t$$ $$x^2+2p+(p/x)^2=t^2$$ $$x^2+(p/x)^2=t^2-2p$$ Because both $x^2,(p/x)^2\ge0$, the whole left side should be $\ge0$. But because it's an equation, this means that…
Kotaka Danski
  • 539
0
votes
1 answer

Expected length of message in communication

The problem goes as follows: A bag contains 16 balls of following colors: 8 red, 4 blue, 2 green, 1 black and 1 white. Alice picks a ball randomly from the bag and messages Bob its color using a string of zeros and ones. She replaces the ball in…
Gokul
  • 514
0
votes
2 answers

Conditional expectation of an unbiased and sufficent statistic

I'm having some trouble understanding the following equality from my course book. Some background: Let $X_1,...,X_n$ be n independent Bernoulli r.v's with unknow parameter $\theta$. $X_1$ is an unibased estimator for $\theta$ and $T(X) =…
Carl Näsvall Sindeby
  • 35
0
votes
1 answer

Set Theory and Expected Value problem

For any subset $S\subseteq\{1,2,\ldots,15\}$, call a number $n$ an anchor for $S$ if $n$ and $n+\#(S)$ are both elements of $S$. For example, $4$ is an anchor of the set $S=\{4,7,14\}$, since $4\in S$ and $4+\#(S) = 4+3 = 7\in S$. Given that $S$ is…
Math_Guy
  • 166
0
votes
1 answer

How to prove that exponential of an expected value of a variable is less than the expected value of the exponential of the variable

I am trying to prove the following: $e^{E(x)} \le E(e^x)$ for a discrete random variable x. I am stuck on how to proceed. None of the usual rules for expected value seem to apply for something like $f(E(x))$. Can I some help? Thanks!!
William Deng
  • 33
0
votes
1 answer

Three questions about the expectation of Brownian Motion.

Let W(r) is a standard brownian motion which follows N(0,r). Then, how to calculate (1) $E(\int^1_0W(r)^3dr)$ (2) $E(\int^1_0W(r)dr)^2$ (3) $E(W(1)\int^1_0W(r)dr)^2$ For (1), my solution is $E(\int^1_0W(r)^3dr)$ = $\int^1_0E(W(r)^3)dr$ =…
Kyoung Hoon Lee
  • 1
0
votes
0 answers

Expected number of moves across a square

Given a square. Each vertex is connected to every other vertex. An ant is located at one of the vertices. There are breadcrumbs on every other vertex. I.e. 3 breadcrumbs, each at rest of the vertices. It will go to every other vertex with equal…
Abhishek Paul
  • 23
  • 3
0
votes
1 answer

Finding the density of $X$

Let $X$ be a random variable with $ E(X^m) = (m+1)! (2^m), \ m=1,2,3, ...$ Find the density of $X$. I tried finding the nth derivative of the moment generating function to solve this question but somehow I can't find the answer.
user90596
  • 129
  • 8
-1
votes
1 answer

Getting $X$ value between two numbers

Let's assume a variable $X$: When we have a value of $100$ -> $X$ will be $50$. When we have a value of $200$ -> $X$ will be $25$. That means $X$ decreased with a steady value a total of $25$ from $100$ to $200$. The question is if the value is…
Makdous
  • 101
-1
votes
1 answer

$E[X^2] = \sum_{x=0}^{\infty}(2x+1)P[X>x]$

How to prove $E[X^2] = \Sigma_{x=0}^{\infty}(2x+1)P[X>x]$? It is easy to use generating function to prove: $E[X]=\sum_{x=0}^{\infty}P[X>x]$. Given that $X$ is discrete random variable and with countable elements.
Clockj
  • 3
-1
votes
1 answer

Probability and statistics expected value

The number of car accidents during one week on a busy motorway is modelled by random variable $X$ with distribution $\mathbb P(X=k) = e^{-\lambda} \cdot \frac{\lambda^k}{k!}$ for $k=0,1,2,\ldots$, where $\lambda=0.5$. How can I solve how many car…
pablo1.2.3
  • 1
-1
votes
1 answer

Prove the expected value of a normal variable

I would like to ask how would I prove that if $X$ is a normal variable with mean $\mu$ and variance $\sigma^2$, then $\mathbb{E}e^{u(X-\mu)} = e^{\frac{1}{2}u^2\sigma^2}$?
Wei Chong
  • 21
  • 4
1 2 3
…
10
11