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.
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Mathematical Expectation. E[E[x]]=E[x]

Is it true that $ E[E[X]]=E[X]$? I can't find this property. If it isn't true, then why $E[(X −E[X])^2]=E[X^2]−E[X]^2$?
Okumo
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$E[1/(M+1)]$ where $M$ is the number of unique items from sampling with replacement

Let $M$ be the number of unique items from sampling $n$ items with replacement from a set of $n$ unique items. Compute the expectation: $$E\left[\frac{1}{M+1}\right]$$ What I've tried It's straightforward to compute the expectation of $M$ by…
dmh
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Does $E[X] \geq E[Y]$ imply $E[\max(a, X)] \geq E[\max(a, Y)]$ for nonnegative variables?

Does $E[X] \geq E[Y]$ imply $E[\max(a, X)] \geq E[\max(a, Y)]$ for nonnegative random variables $X$ and $Y$ and a constant $a >0$? Here is what I have tried: We know $E[\max(a, X)] \geq E[X]$ and similar for $Y$. This says that $E[X]$ and $E[Y]$ are…
Max
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Expected value E[7Y] from joint density function

So I'm trying to solve this problem. Random vector $(X,Y)$ has the joint density function given by $f(x,y)=6 x y^2$, when $0
mrrobot
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On average, how many random real numbers from $0$ to $1$ are required so that their ALTERNATING sum exceeds $1$?

You have a random number generator that generates random uniformly distributed real numbers from $0$ to $1$. Generate a number, then subtract a new number, then add a new number, then subtract a new number, and so on. On average, how many numbers…
Dan
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expected value rule of Uniform random variable

Let be a uniform random variable on the range $\{−1,0,1,2\}$. Let $=^4$. Use the expected value rule to calculate $\mathbb{E}[Y]$. Why isn’t the answer $1/4 \cdot(-1)^4 +1/4\cdot (0)^4+ 1/4 \cdot(1)^4 + 1/4 \cdot(2)^4$
Laura Stewart
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What is the complex conjugate?

I was doing a problem today in the book that deals with Schrodinger's equation for a potential well. Basically a potential well is when the potential energy is $0$ for some interval and then $V_1$ everywhere else, which is used for trapping…
Jonathan Aguilera
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Clarify my Thinking about expected values and coin flips.

Problem: Find The expected number of coin flips until two heads in a row or three tails in a row occurs. Answer: $\frac{21}{5}$. This problem is not hard to just manually solve using states and creating a system of equations—I’ve already done this…
solasky
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Expected longest run of consecutive heads or tails in $N$ tosses

Suppose you toss a coin $N$ times. What is the expected longest run of consecutive heads or tails in $N$ tosses? e.g., suppose $N = 5$. If the sequence is HHTTT, then the longest length here is 3. If the sequence is HTHHT, then the longest length is…
24n8
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Expected Value of $\exp\left[{a+bX+X'cX}\right]$

Is there a closed-form expression for $$ \mathbb{E}\left[e^{a+bX+X'cX}\right], $$ where $a$ is a constant, $b$ is a vector of constants and $c$ a matrix of constants. $X$ is a vector of Gaussian RV's with vector of means $\mu$ and variance…
Breugem
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The expectation of an expectation

This may seem trivial but just to confirm, as the expected value is a constant, this implies that the expectation of an expectation is just itself. It would be useful to know if this assumption is correct or if any subtleties cause this not to be…
Rosco
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Finding expectation via indicator variables

This is not homework. I am not a student. Consider a sequence of independent tosses of a biased coin at times k=0,1,2,…,n. On each toss, the probability of Heads is $p$, and the probability of Tails is $1−p$ . A reward of one unit is given at time…
Abhi Minhas
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Expected value problem with unintuitive result

Suppose you draw $n$ samples from a uniform distribution where $X$ is from $(-100,100)$ Call your samples $X_1, X_2, . . . X_n$. Next you draw one more sample, call it $X_0$. Then you start comparing $X_0$ to each of the $X_1, X_2, … X_n$ in turn,…
Sam
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Branching process: Evaluating the probability of extinction

Let $Z_n$ be a branching process, with $Z_0 = 1$. Let the Progeny N be distributed as $P(N=0)=P(N=2)=\frac{1}{2}$. I want to solve the following problems: 1) Show that $E[Z_{n}] \to 1$, as $n \to \infty $ 2) Show that $Z_n$ cannot converge in…
Carl Näsvall Sindeby
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Cutting a cake n-times - expected number of pieces

I'm trying to solve a brainteaser: A 2D circle (cake) is cut randomly with straight lines n-times. What's the expected number of pieces? Could you give me a hint on how to approach this? Is it possible to solve it analytically or by…
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