Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [expected-value]

Questions about the expected value of a random variable.

0
votes
0answers
14 views

Expectation and Variance of Infinite Sequence

We have an infinite sequence of independent variables where we take k labelled balls and put them into k labelled slots so that there is one ball per slot. We want to find the expected value and ...
0
votes
1answer
15 views

Does anything like expectation of joint distribution exist?

I know how to find the expectation of a function of a Random Variables, I was just wondering that does expectation of a joint distribution exists? I think, since expectation is an average which by ...
0
votes
0answers
24 views

Expected Value in joint probability distributions

I am very puzzled as to how they arrived at μ=2/5 in this example. I understand that μ is just expected value which would mean summing x or y with its possible realizations, but if x and y are max. 1, ...
-4
votes
0answers
36 views

Expectation probability [on hold]

In a statistics examination, the mean score for 400 students is 78% and the standard deviation is 10%. How many students have scores (a) (i) more than 70 (ii) less than 90% (iii) between 70% and 90% ?...
0
votes
2answers
46 views

Hard expected value problem

Given a series of discrete random variables $Y_2, Y_3...Y_n$, such that for all $Y_i$ : $P(Y_i = e^i) = \frac{1}{i}$, $P(Y_i = 3) = \frac{1}{3}$, $P(Y_i = X) = \frac{2}{3} - \frac{1}{i}$, and $X$ ...
0
votes
0answers
20 views

Mathematical derivation of why Bagging reduces variance

I am having a problem understanding the following math in derivation that bagging reduces variance. The math is shown but can not work it out as some steps is missing. link
0
votes
0answers
38 views

Conditional Expectation Inequality for bounded moment

I was going through a proof and encountered the following: Let $X$ be a random variable with $\mathbf{E}[X] = \mu$ and $\mathbf{E}[(X-\mu)^2] = \sigma^2$ and let $\mathbf{E}[(X-\mu)^4] \leq C_4 \...
-1
votes
0answers
21 views

Probability, expectations [on hold]

i need help with this exercise. (I google translated from norwegian so sorry if the english is bad) We want to find the proportion of the working population that is unemployed at some point. A ...
0
votes
0answers
18 views

Expected value for the eigenvalue of random orthogonal matrix

I am interested in finding the expected value of the product of two eigenvalue with respected to the GOE ensemble. So I have a random $n\times n$ matrix with all off diagonal elements $\mathcal{N}(0,...
0
votes
1answer
28 views

Sucker Bet - Coin Flipping Stochastic Process

Having a lot of trouble working out this exercise. I have tried constructing the 8x8 matrix with all possible combinations of three flips of the coin {HHH, HHT, HTH, ... , TTT} and then calculating an ...
1
vote
2answers
20 views

Expectation of lost ball number

I have expectation problem that sounds like this: In the box we have 5 balls numbered U={2,2,2,3,3}. We lost one ball and we don't know which one. Then one ball was taken out from the box which ...
1
vote
1answer
32 views

1st Yr Probability: what is wrong with my approach? (using indicator RVs)

Joe’s iPod has $500$ different songs, consisting of $50$ albums of $10$ songs each. He listens to $11$ random songs on his iPod, with all songs equally likely and chosen independently (so repetitions ...
0
votes
0answers
23 views

Can we find the expected value of $Z$, where $Z=X^p$, and $X\sim\mathcal{N}(\mu_X,\sigma_X)$?

Can we find the expected value of $Z$, where $Z=X^p$ with $p>0$ and $X\sim\mathcal{N}(\mu_X,\sigma_X)$ ? With the help of Gaussian distribution raised to a power, I think the pdf of Z should be, $...
-2
votes
1answer
36 views

If E[ln(Y)] = 0, how can I find the value of E[Y]? [on hold]

I would like to know if it is possible to know anything about E[Y]
1
vote
2answers
62 views

is this true: Var(|X|) ≤ Var(X)? [on hold]

I am trying to decide whether $$\text{Var} (\lvert X \rvert) \le \text{Var}(X)$$ is true. I am getting stuck because I don't know if $$E^{2}[ \lvert X \rvert ] \gt E^{2} [ X ]$$
-4
votes
1answer
36 views

Why is $E[(X − b) ^2 ]$ minimal when $b = µ$? [on hold]

If $X$ is a random variable with a mean µ and a variance $σ^2$, why is $$E[(X − b) ^2 ]$$ minimal when $b = µ$?
0
votes
1answer
12 views

Find values of $a$ and $\lambda$ for which $Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale

Find values of $a$ and $\lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,b\in\mathbb{R}$ can be positive as well as negative, since $...
0
votes
1answer
34 views

Conditional probability and expected value calculation

A football team, LIBO, wins a match with probability 0.75 irrespective of its opponents. What is the probability that the team wins 4 matches out of 5 matches? In a knockout tournament, LIBO ...
0
votes
0answers
17 views

Laplace functional for given process.

Consider $X_{1} \dots $ i.r.v with $Exp(\lambda)$ distribution. Let's consider $S_n = X_{1} + \dots + X_{n}$. So we may construct a point process $Y = \{Y(B) = \sum \mathbb{1}_{B}(S_n) , B\in \...
0
votes
0answers
20 views

i.i.d. estimation of variance

Let ${X_i}$ for $i\in N$ be a string i.i.d. with a known expected value of $\mu = E (X_1)$. Investigate the properties the estimator of the variance given by the formula : $\frac{1}{n} \sum(X_i ^2 - \...
1
vote
1answer
9 views

Variance of combination of Brownian Motions

Let $Z(t)=W(t)-\frac{t}{T}W(T-t)$ for any $0\leq t\leq T$ with $W(t)$ a Brownian motion, find the variance of $Z(t)$. My attempt: $Var(Z(t))=\mathbb{E}(Z(t)^{2})-\mathbb{E}(Z(t))^{2}$ $Z(t)=W(t)-\...
1
vote
1answer
56 views

Prove of $E\left(|X+Y|^a\right)\ge E\left(|Y|^a\right)$?

Let $E(V)$ be the expectation of $V$. It is also known that $E(X)=0, a>1, E\left(|X|^a\right) < +\infty$ and $E\left(|Y|^a\right)< +\infty$. $X$ and $Y$ are independent. How can I prove that $...
-1
votes
0answers
16 views

Reference for EV/ betting game questions. [closed]

Need some good material to practice questions on probabilistic games. Mainly questions on the lines of finding EV and deciding whether it's a fair bet or not, or how much should one bet.
2
votes
0answers
25 views

Have I been rigorous in this simple argument with summation and expectation operators?

I am told that $A$ and $B$ are random variables, and that $\mathbf{E}(A|B) = \gamma B$. Define $D = A/B$. Using the law of iterated expectations it can be shown that $\mathbf{E}(D) = \gamma$. Now ...
2
votes
1answer
58 views

$\Bbb E(X\mid X^2)=X$

Assume $X$ is exponential distribution. Is the following proof correct? $$\Bbb E(X\mid X^2)=X$$ This holds because $X$ is measurable in $\sigma\langle X^2\rangle$$\left(X = \sqrt{X^2}\text{ because ...
0
votes
0answers
13 views

Expected value of $(z- \langle z \rangle )^2$ in a potential well

today I am asking for help in checking my work. It seems like I am off by a constant for calculating the the expected value of $(z- \langle z \rangle )^2$ in a potential well with interval $\frac{-n \...
0
votes
0answers
58 views

Expectation of the Error

Suppose $X_1,\ldots,X_n, Y_1,\ldots, Y_n,U_1,U_2,V_1,V_2$ are i.i.d. uniformly random variables in $[-1,1]$. Let $M_1 = \frac{V_2-V_1}{U_2-U_1}$ and $B_1 = V_1-U_1M_1$. Let $M_2,B_2$ be the ...
0
votes
1answer
34 views

are integration limit to find expected value from a pdf inclusive (or exclusive)?

This is my probability density function (pdf) $$pdf = e^{-\frac{r}{\lambda}} \frac{1}{\lambda}$$ I want to find the expected value (EV) from $0<r \leq r_0$, $r$ is the random variable. My first ...
1
vote
0answers
12 views

is $E[W\mid X]P[X]$ the same as $E[W\Bbb 1\{X\text{ takes place}\}]$

In a proof of I am reading there is the following setting: $W$ is a random variable such that $ E[\mid W\mid]<\infty$ and $T$ is another variable (a stopping time) that takes values in $\Bbb N\cup\...
0
votes
0answers
29 views

Convergence in probability of non-negative random variable and expectation

Let $\mathbb E X$ exists and $X_n : X_n$ non-negative and $X_n \to^P X$ . How to show, that $\mathbb E X_n \to \mathbb E X \iff \mathbb E |X_n - X| \to 0$?
0
votes
1answer
13 views

Covariance of centred random vairables

Let $X$ and $Y$ be two random variables, with $\mathbb{E}(X)=\mathbb{E}(Y)=0$. Is it true that $\mathbb{E}(XY)=0$ ? In other word, is it true that cov$(X,Y)=0$ ?
0
votes
0answers
45 views

Expected value questions on games! [closed]

I need material to practice EV games questions.But I lack practice in betting questions where a set-up of a game is given and one has to respond to the best strategy or best bet to take. A good book ...
0
votes
1answer
26 views

How can $\sigma^2$ be derived as a function of $\mu$ in a Gaussian pdf?

I have a Gaussian pdf defined as $$f_X(x) =\frac{1}{\sigma\sqrt{2\pi}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$$ whose $\mu = \frac{d^2}{6D}$, where $d$ is distance parameter and $D$ is the ...
0
votes
1answer
26 views

Is the expectation of the derivative of the distribution function always 0?

I was wondering if the expectation with respect to an r.v. of the pdf of that r.v. is always 0. I came to this conjecture by accident, and then tried it out by using pdfs composed of different ...
1
vote
0answers
15 views

Basic Question on Conditional Expectation (conditioning with different sub-$\sigma$-algebras)

I have a very basic question concerning conditional expectation. Let $(\Omega, \Sigma, p)$ be a (finite) probability space and let $\Sigma_k$ be a sub-$\sigma$-algebra (with $k \in K := \{1,2\}$), ...
0
votes
0answers
10 views

Upper bound on the error when approximating expectation of a function of a random variable using Taylor series.

If I have a non-negative random variable $X$ whose distribution is not known but moments are known and I want to find the expected value of a function of this random variable, say, $f(X) = (1/(1+X)) + ...
0
votes
0answers
23 views

Expected value of a minimum and maximum of a collection of independent random variables

Suppose I have two independent random variables $X$ and $Y$ with probability density functions $f_X(x)$, where $0 \leq x \lt a~$, and $f_Y(y)$ with $0 \leq y \lt b$. Let $T = \min\{X, Y\}$ and $W =...
0
votes
1answer
24 views

The expected value within a time interval

Let's say that I have a group of 71 people and I'm trying to find the expected value of how many of their birthdays fall within a specific 6 day time interval (June 12-17). Would this type of ...
0
votes
1answer
39 views

Proving that $\Gamma(\frac{1}{2})=\sqrt(\pi)$ using the expected value of standard normal variable (integral calculation)

I'm looking to prove that $\gamma$$(\frac{1}{2})=\sqrt(\pi)$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt(2\pi)}e^{\frac{-z^2}{2}} z^2 dz$ (where $Z$ is a standard normal ...
1
vote
2answers
41 views

Expected value of mean plus the deviation from the mean

Suppose $r$ and $\theta$ are random variables with: $r = \bar{r} + \tilde{r}$ $\theta = \bar{\theta} + \tilde{\theta}$ where $\bar{r}, \bar{\theta}$ are the means, and $\tilde{r},\tilde{\theta}$ ...
1
vote
2answers
38 views

Problem of probability distribution

A box contains $N$ tickets numbered $1, 2, 3,...., N$. If $m$ tickets are drawn one by one from the box without replacement, then find the mean of the sum of the numbers obtained on the tickets drawn. ...
0
votes
0answers
24 views

Expected value and Variance question

So let $A$ be the event the die falls either "one", "two", or "three" Let $B$ be the event the die falls either "four" or "five" and let $C$ be the event the die falls "six" $P(A) = \frac{1}{2}$, $...
0
votes
1answer
31 views

Is this a partition of a sample space?

Suppose I have a random variable $ X: \Omega \longrightarrow R$. Where $\Omega$ is the sample space. Suppose then that I have another random variable $Y$ which is some function of $X$, $Y=f(X)$. ...
0
votes
1answer
29 views

Expected number of colors in a sampling of colored balls without replacement,

Suppose there is a box containing differently colored balls. There are $G$ colors, each color having $n$ balls of this color, i.e. $G \times n$ balls. What is the expected number of different colors ...
3
votes
0answers
46 views

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 $V_1$ ...
0
votes
0answers
22 views

Can anyone support me with this proof of Moran's I expectation?

I have read some articles about Moran's I statistic and I couldn't found the proof of its expectation. Then I want to prove it. The Moran's I is, $$ I=\frac{N}{W}\frac{\sum_{i=1}^{N} \sum_{j=1}^{N} ...
0
votes
1answer
25 views

Is expected value $E[X_1;X_1\leq X_2]= E[X_1;X_1< X_2]$

Let $X_1, X_2$ two independent random variables with PDF $f_{X_i}(x_i)$. Is this formula is true $$E[X_1;X_1\leq X_2]= \int_{x_2=0}^{\infty}\Big(\int_{x_1=0}^{x_2}x_1 f_{X_1}(x_1)d x_1\Big)f_{X_2}(x_2)...
0
votes
1answer
29 views

Convergence in probability and expectation

Let $\mathbb EX$ exist and $X_n$ converges in probability to $X$. How to prove that $\mathbb EX_n → \mathbb EX$ then and only when $\mathbb E| X_n - X | → 0$?
0
votes
1answer
27 views

How was the expected value of this derived

I need help with understanding how the expected value was derived for this question: I know how to calculate the expected value for when $\theta=0$ and $\theta=1$, but in the solutions, the expected ...
0
votes
1answer
24 views

Covariance of square root for two bins of a multinomial

Take $(X_1, \dots, X_k) \sim Multinomial(n, (p_1, \dots, p_k))$. Do we have a closed form expression for $\mathbb{E}[\sqrt{X_i X_j}], i\neq j$ ?