What is Expected value: Definition and 250 Discussions

In probability theory, the expected value of a random variable


{\displaystyle X}
, denoted



{\displaystyle \operatorname {E} (X)}



{\displaystyle \operatorname {E} [X]}
, is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of


{\displaystyle X}
. The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. Expected value is a key concept in economics, finance, and many other subjects.
By definition, the expected value of a constant random variable


{\displaystyle X=c}


{\displaystyle c}
. The expected value of a random variable


{\displaystyle X}
with equiprobable outcomes









{\displaystyle \{c_{1},\ldots ,c_{n}\}}
is defined as the arithmetic mean of the terms




{\displaystyle c_{i}.}
If some of the probabilities






{\displaystyle \Pr \,(X=c_{i})}
of an individual outcome



{\displaystyle c_{i}}
are unequal, then the expected value is defined to be the probability-weighted average of the



{\displaystyle c_{i}}
s, that is, the sum of the


{\displaystyle n}








{\displaystyle c_{i}\cdot \Pr \,(X=c_{i})}
. The expected value of a general random variable involves integration in the sense of Lebesgue.

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  1. Kyuubi

    Expected value of momentum P in terms of k

    Now if I'm given a ##\phi(k)##, and I'm asked to find ##\langle p \rangle##, ##\langle p^2 \rangle##, etc. Am I justified to say that ##\langle p \rangle = \hbar \langle k \rangle## and that ##\langle p^2 \rangle = \hbar^2 \langle k^2 \rangle## ?
  2. A

    Why is the work done double its expected value? (conveyer belt)

    The question was this: My calculations show that the answer should be equal to work done on crate to make it reach the same velocity which is equal to 216 J but the answer given is 432 J It is believed that extra energy is needed to overcome friction but friction is an internal force and...
  3. WMDhamnekar

    I Expected number of random variables that must be observed

    In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ## In answer to (b), XN is wrong. It should be XN=p-4q-3 - p-3 q-2- p-2 q-1 - p-1. This might be a typographical error. Is my answer to (a) correct?
  4. graviton_10

    Expected value of variance of Hamiltonian in coherent states

    I am trying to find the expected value of the variance of energy in coherent states. But since the lowering and raising operators are non-hermitian and non-commutative, I am not sure if I am doing it right. I'm pretty sure my <H>2 calculation is right, but I'm not sure about <H2> calculation...
  5. H

    Difference between average position of electron and average separation

    Hi, I asked this question elsewhere, but I didn't understand the answer. It seems to be easy to understand, but for some reason I'm really confuse. I'm not sure how to find the average position of an electron and the average separation of an electron and his proton in a hydrogen atom. To be...
  6. C

    Comp Sci Expected number of collisions for hashing with chaining

    Attempt: Denote ## T## as the hash table, ## h ## as the hash function. Denote ## n ## as the number of keys in the universe, and ## m ## as the size of hash table. Order the set of keys from the universe as ## \{ k_1 ,..., k_n \} ## such that ## k_i \leq k_j ## where ## i \leq j ##. Note...
  7. Euge

    POTW Expected Value of a Maximum

    If X and Y are independent standard Gaussian variables, find the expected value of the maximum of X and Y. Note: This problem is due to @Office_Shredder.
  8. Feynstein100

    I Is there a way to calculate expected value from probabilistic data?

    So I ran a python simulation of 1,000 games of toss (50/50 odds) where each game consists of 100,000 consecutive flips. The result was this: 1000 is our starting balance and as expected, there's a nice normal distribution around it. I also calculated the average value after all the games and it...
  9. V

    Expected Value of Election Results

    I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong? Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house. There are 4 cases: winning the senate and house (##S \cap H##), winning just...
  10. Bruno Cardin

    A Expectation value in Heisenberg picture: creation and annihilation

    So, I have a hamiltonian for screening effect, written like: $$ H=\sum_{k}^{}\epsilon_{k}c_{k}^{\dagger}c_{k}+ \frac{1}{\Omega}\sum_{k,q}^{}V(q,t)c_{k+q}^{\dagger}c_{k} $$ And I have to find an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}##. I...
  11. WMDhamnekar

    Using Multiple integrals to compute expected value

    I want to know how did author derive the red underlined term in the following Example?
  12. WMDhamnekar

    MHB Probability, Expected value, joint P.D.F. and order statistics

    I want to know how did author derive the red underlined term in the below given Example? Would any member of Math help board enlighten me in this regard? Any math help will be accepted.
  13. M

    Python Programming a recursive expected value card game

    Hi PF! Take a deck of cards. 26 are red and 26 are black. You draw cards randomly without replacement: if the card is red you gain a dollar, if black you lose a dollar. What's the expected value? My thought process was to look at simple cases and build to a 52 card deck. We know if ##r=0##...
  14. docnet

    Find the expected value of a coin flipping game

    summary: you toss six coins. if you have 3 heads, you don't get any money. if you have 4 or more heads, you get the number of heads amount of dollars. if you have 2 or less heads, you toss six coins again and get the number of heads amount of dollars. the number of ways to get 0 head...
  15. mcas

    Show that an expected value of a vacuum state is equal to 1

    \langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| - \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow} \hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| \hat{c}_{k...
  16. M

    Expected value of median of rolling three fair dice

    Hi, I was reading this problem and I found a solution on Math Stackexchange which I don't quite understand. Question: Calculate the expected value of the median of rolling a die three times. Attempt: I read the following answer on math stack exchange here "As already noted in a comment, the...
  17. R

    Expected value question -- Probabilities when tossing two coins

    Summary:: checking an expected error Given the question: "If a person tosses two coins and gets two heads, the person wins $10. How much should the person pay if the game is to be fair?" The book gives the answer as $2.5 while I calculate $3.333... E(X) = 0 = $10(1/4) - a(3/4) => a =...
  18. K

    I How do I find the expected value and median of a probability density function?

    Hey everyone, I have been struggling to find the expected value and median of f(x) = 1/2e^-x/2, for x greater than 0. I am just wondering how I do so? Thank you.
  19. J

    How to find an expected value for magnetic field strength?

    Our class modified an experiment to measure the magnetic field strength in mT between 5cm and 30cm, and I have plotted data and found that the relationship resembles a power relationship (using a log vs log graph). In order to find the percentage uncertainty for the whole experiment I need the...
  20. L

    MHB Asian Zing Wings: Calculate Expected Value

    Calculate the expected value for the number of Asian Zing wings sold
  21. M

    MHB Distribution, expected value, variance, covariance and correlation

    Hey! :giggle: Let $X$, $Y$ and $Z$ be independent random variables. Let $X$ be Bernoulli distributed on $\{0,1\}$ with success parameter $p_0$ and let $Y$ be Poisson distributed with parameter $\lambda$ and let $Z$ be Poisson distributed with parameter $\mu$. (a) Calculate the distribution...
  22. U

    MHB How to find PDF and Expected value of max(x,0), for a random variable x

    Let $a,b,c, \tau$ be positive constants and $x$ is an exponentially distributed variable with parameter $\lambda = 1$, i.e. $x\sim\exp(1)$. \begin{equation} E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+ \end{equation} where $[z]^+ = \max(z,0)$ How can I find The PDF for $E$ The...
  23. ?

    B Solving a RNG Problem: What is the Expected Value of N?

    Hey all, So this time I have a different kind of question - namely, "what is this called?" I recall hearing/reading this in at least two places, one of which was YouTube. The idea is the following: A RNG picks an integer uniformly from 1 to N. It picks 4. What is the expected value of N? I'm...
  24. Adams2020

    A Higgs particle and non-zero expected value in vacuum

    What does the phrase “Due to Lorentz invariance, only the Higgs particle can have a non-zero expected value in a vacuum” mean?
  25. J

    I Expected value of a variable and its reciprocal

    If E(X1)>E(X2), does it mean that E(1/X1)<E(1/X2)? Thanks!
  26. D

    Expected value of two uniformly distributed random variables

    ##X_1## and## X_2## are uniformly distributed random variables with parameters ##(0,1)## then: ##E \left[ min \left\{ X_1 , X_2 \right\} \right] = ## what should I do with that min?
  27. M

    Expected Value: Bidding for a House

    Summary:: The price of a house is uniformly distributed between 0 and 1000 but we do not know its exact value. If we place a bid higher than the value, then we obtain the house, but if our bid is lower then we get nothing. If we know we can sell the house on to another person (guaranteed) for...
  28. Addez123

    Calculate the expected value for filling up a cup

    My guess is that g(x) = x? The limits of integration should be 0 to a, since after a the cup flows over. If I put these in, I get the solution (I've doubled checked with wolfram alpha that it's correct): $$E(Y) = ln(a+1) + 1/(a+1) - 1$$ The textbook solution is just ln(a+1). I'm super new to...
  29. Addez123

    Calculating the expected value of a dice roll

    I write p(k) as: $$p(k) = 1/6, k = 2,3,4,5$$ $$p(k) = 2/6, k = 6$$ Is that wrong? Because then the expected value becomes $$1/6 * 4 + 2/6 * 6 = 8/3$$ While my book says 11/3
  30. WMDhamnekar

    MHB Expected value and equality to sums

    How to show that$E[N]=\displaystyle\sum_{k=1}^\infty P{\{N\geq k\}}=\displaystyle\sum_{k=0}^\infty P{\{N>k\}}$ If any member here knows the answer, may reply to this question.:confused:
  31. I

    MHB Expected Value of Cup of Coffee in Flip a Lid Contest

    In its flip a lid contest, a coffee chain offers prizes of 50,000 free coffees, each worth \$1.50; two new TVs, each worth \$1200; a snowmobile worth \$15 000; and sports car worth \$35 000. A total of 1 000 000 promotional coffee cups have been printed for contest. Coffee sells for \$1.50 per...
  32. karush

    MHB C Expected value for the total number of points awarded on any toss of coins

    https://www.physicsforums.com/attachments/9838 well not sure why we need 3 different coins other than confusion also each toss at least 2 coins have to have the same face frankly not sure how any of these choices work didn't want to surf online better to stumble thru it here and learn it better...
  33. cemtu

    Quantum Mechanics Hydrogen Atom Expectation Value Problem

    I can not solve this problem: However, I have a similar problem with proper solution: Can you please guide me to solve my question? I am not being able to relate Y R (from first question) and U (from second question), and solve the question at the top above...
  34. rafgger

    A Expectation of a Fraction of Gaussian Hypergeometric Functions

    I am looking for the expectation of a fraction of Gauss hypergeometric functions. $$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$ Are there any identities that could be used to simplify or...
  35. RealKiller69

    Why Does My Calculation Have an Extra Factor of 2?

    My solution is not the same, it has an additional *2 factor and i can't see any mistakes.
  36. peace

    I The expected value (or mean) of X

    hi. How to obtain the expected value of x in the momentum space ?
  37. J

    I Calculating the expected value of the square of an integral of Brownian Motion

    For a standard one-dimensional Brownian motion W(t), calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$I can't figure out how the middle term simplifies. $$ \mathsf E\left(\int_0^T W_t\mathrm dt\right)^2 = \mathsf E\left[T^2W_T^2\right] - 2T\mathsf E\left[W_T\int_0^T...
  38. C

    I Expected value of 2^X and 2^-X

    For the following distributions find $$E[2^X]$$ and $$E[2^{-X}]$$ if finite. In each case,clearly state for what values of the parameter the expectation is finite. (a) $$X\sim Geom(p)$$ (b) $$X\sim Pois(\lambda)$$ My attempt: Using LOTUS and $$E[X]=\sum_{k=0}^{\infty}kP(X=k)=\frac{1-p}{p}$$...
  39. T

    MHB Calc Expected Value & Variance of Multivar. Func.

    Hey, I've got this problem that I've been trying to crack for a while. I can't find any info for multi-variable expected values in my textbook, and I couldn't find a lot of stuff that made sense to me online. Here's the problem. Find $E(C)$ Find $Var(C)$ I tried to get the limits from the...
  40. J

    Finding the expected value of position in a Potential Well

    Homework Statement Hello today I am solving a problem where an electron is trapped in a potential well. I have a solved Schrodinger's Equation. I am having problems in figuring out what the wave function should be. When I solved the equation I got a complex exponential. I know I cannot use the...
  41. pawlo392

    Calculating Expected Value and Variance of Coin Toss Results

    A coin had tossed three times. Let ##X##- number of tails and ##Y##- number of heads. Find the expected value and variance ##Z=XY##. My solution: We know, that ##Y=3-X##, so ##Z=(3-X)X## for ##X=0,1,2,3##. ##Z=2## for ##X=1,2## and ##Z=0## for ##X=3,0## So, ##E(Z)=E((3-X)X))= 2 \cdot ⅜ +2 \cdot...
  42. A

    The expected value for 7 heads in a row

    Homework Statement We have a coin with probability ##0\leqslant p \leqslant 1## of getting heads. We flip the coin until we get ##7## heads in a row. Let ##N_7## be the number of necessary flips to get the ##7## heads in a row. What is the expected value ##E(N_7)##? Homework Equations The...
  43. A

    Expected value of a function of a random variable

    Homework Statement Let X be a random variable. It is not specified if it is continuous or discrete. Let g(x) alway positive and strictly increasing. Deduce this inequality: $$P(X\geqslant x) \leqslant \frac{Eg(X)}{g(x)} \: $$ where x is real. Homework Equations I know that if X is discrete...
  44. benorin

    I Calculating Average Value of Damage w/ 30% Crit Chance

    How do I calculate the average value of some damage values where there is say a 30% chance of critical hit and on a crit 150% damage is done? I run into this problem often, if I had to guess I'd say it's likely related to the binomial distribution because there's either a crit or not, fixed...
  45. S

    Calculating the expected value for a probability

    Homework Statement An ice-cream store makes 150 ice-cream balls every day. The cost of making each ice-cream ball is $3. The price of an ice-cream ball is $8. The demand distribution is as follows: 100 ice-cream balls with probability 25%, 150 ice-cream balls with probability 50%, and 200...
  46. M

    MHB How many steps are needed to reach 12 from 0 in a probabilistic sequence?

    Hello, I would like to ask, if I am right with my computation. Let´s have a set of integers from 0 to 12. We start at 0 and we can go to 1 with the probability 1. From 1 we can go to 1 or back to 0, both with probability 0.5. When we start at zero, how many steps (exp. number of them) do we...
  47. H

    Expected value of binomial distribution

    Homework Statement A random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a uniform(0,1) random variable. What is E[Y]? Homework Equations E[Y] = np The Attempt at a Solution The key is determining the probability of...
  48. P

    Most likely location of a particle versus expected value of

    Hello. I'm using Griffiths' introductory textbook on quantum mechanics, which I have just begun, and have arrived at a question on a simple wave function: The answer to the first question I know to be √(3/b). The answer to part e is (2a + b) / 4. My problem is that I don't understand how...
  49. M

    MHB How to calculate $E(X_iX_j)$ with $i\ne j$?

    Hey! :o The random variables $X_1, X_2, \ldots , X_{10}$ are independent and have the same distribution function and each of them gets exactly the values $\pm 2$ and with equal probability. We define the random variable $S=X_1+X_2+\ldots +X_{10}$. I want to calculate $\mathbb{E}(S^2)$...
  50. M

    MHB Greatest probability - Expected value

    Hey! :o The geometric distribution with parameter $p\in (0,1)$ has the probability function \begin{equation*}f_X(x)=p(1-p)^{x-1}, \ \ x=1, 2, 3, \ldots\end{equation*} I have shown that $f_X$ for each value of $p\in (0,1)$ is strictly monotone decreasing, as follows...