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Questions tagged [expected-value]

Questions about the expected value of a random variable.

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ls it possible to construct discrete r.v.s given expectation and variance?

Suppose there is a discrete r.v.s X, all we know is: E(X) = 10 and VAR(X) = 2500 Any general way to find PMF of X? Thanks.
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How to determine $E[X^3]$ given $E[X^2]$ and $E[X]$

Let $X$ be a discrete random variable with $E[X] = 2$ and $E[X^2] = 4$. Find $E[(2 + X)^3]$. When I proceed to $E[2+4X+8X^2+X^3]$ I don't know how to calculate $E[X^3]$.
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If distribution's mean is increasing, Is positive increasing function's expectation increasing?

Let me assume that the variable $x\in [a,b]$ and $y\in[a,b]$ where $a,b (a<b)$ are real numbers. I have positive increasing function $g:[a,b] \rightarrow [c,d]$, where $c,d(c<d)$ are real ...
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expectation of a linear operator

Could anyone help me with this? We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$ for a system $X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots,$ $\...
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What is the expected value: [on hold]

How would I calculate the expected value of: There is a game involving opening doors. There are 10 doors and 3 contain normal balls while one contains a gold ball. One gold ball is worth 3 points ...
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1answer
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Questions of Variance, mean and interpretation

Let X be a continuous random variable with the density function: $f_x(x) = \begin{cases} x+1, & \text{if}\ -1\leq x \leq 0 \\ -x +1, & \text{if}\ 0 \leq x < 1\\ 0 &...
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What should be a minimal stake for Casino to secure the profit?

Let's assume Casino has a game, where the expected value of an outcome is 60% for Casino. The casino has 1000 units of money in reserve for this game. Stakes are always 500 units of money. Now John ...
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Finding the expected number of flips needed for a coin having probability $p$ of landing on heads

A coin, having probability $p$ of landing on heads and probability of $q=1-p$ of landing on tails. It is continuously flipped until at least one head and one tail have been flipped. a) Find the ...
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1answer
27 views

Compute moments of Brownian motion stopped at exit time of $[a,b]$

Given $B_t$ a standard brownian motion and $a < 0 < b$ Set $T = \inf\{ t : B_t = a \vee B_t = b\} $ For any $\alpha \in \mathbb{Z}^+$, find $EB_T^\alpha$. I know I can use optional ...
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Conditional expectation problem with integration limit and not only

We are in the shopping center. Customer is paying with card with chance $60\%$, and witch cash with probability $40\%$. When client is paying with card the time of service has exponential distribution ...
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1answer
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“Normalized” covariance matrix of a Gaussian random vector

Let $X\sim\mathcal{N}(0,I_{d})$. I would like to compute the the following quantity: \begin{equation} \mathbb{E}\bigg[\frac{XX^{\top}}{\|X\|_{2}^{2}}\bigg]. \end{equation} Letting $B=\frac{XX^{\top}}{...
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1answer
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How to compute a conditional expectation

I want to compute a conditionnal expectation, i know that $Z=(Z_1,\ldots,Z_p)'$ where $ Z_j=\Phi ^{-1}(U_j)$ with $Z \sim N(0,R(\theta))$ and $R(\theta)$ the $p \times p$ positive definite ...
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How to define a function whose expectation operator changes with a variable?

To elaborate, I have a function, $C\left(a,b,r\right)$, where, $a,b \in \left[0,1\right]$ and $r \in \left[r_0,r_1,r_2,r_3\right]$. Also, $C_{0,0} = \displaystyle{\min_{r}}~ C\left(0,0,r\right)$, ...
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Gaussian mixture model: sum of 2 random variables [on hold]

We have data for $X$ (integers). Model $X = Y+Z$. Where $Y\sim N(\mu_1,\sigma_1)$ and $Z\sim N(\mu_2,\sigma_2)$ are latent variables. How to use a Gaussian mixture model or EM algorithm to estimate:...
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Intuition behind expected value formula? [duplicate]

I just learned the expected value formula for a positive random variable: $E[X] = \int_0^\infty P(X>x)dx$. I can follow the proof but I'm a little unclear as to what the intuition behind this ...
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Is there some sort of a closed formula for $\alpha_n$?

Suppose $\{X_i\}_{i = 0}^\infty$ is a Galton-Watson branching process ($\{X_i\}_{i = 0}^\infty$ is a sequence of random variables satisfying the conditions $P(X_0)= 1$ and $P(X_i) = \Sigma_{i = 1}^{X_{...
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2answers
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Expected number of turns so that all balls get painted in blue color

A container contains $5$ red balls. On each turn,one of the balls is selected at random,painted blue,and returned to the container.The expected number of turns it will take before all $5$ balls are ...
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1answer
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Expectation of a hitting time

I'm trying to find the expectation of a stopping time. Specifically, Let $T_1,...,T_n$ be i.i.d exponential random variables with mean $1$. Let $S_n = T_1 + ... + T_n$ denote their partial sum. ...
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1answer
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Proof with the expected value [on hold]

I wonder if the following statement is true: Assuming that $X>Y$ with probability 1, show that $$ E|X-Y| = E(X-Y) = EX - EY. $$ Intuitively I think it is correct, however I need a formal proof.
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Gambler's ruin vs point difference

Two people want to investigate who is best at playing Scrabble. They decide to settle this by playing a series of games. Assume that there is always a winner for each game, and that winning or losing ...
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3answers
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What is that inequality - Cauchy-Schwarz for a single random variable?

In the following article: https://link.springer.com/content/pdf/10.1023%2FA%3A1018054314350.pdf below equation (4.1) there is a statement that: $$EZ^2 \geq (EZ)^2$$, where $Z$ is a random variable. ...
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1answer
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Moment estimator $\hat{\theta}$ of $\mathrm{Beta}(\theta,1)$ and bias of $\hat{\theta}$

I'm trying to find the moment estimator for the density function $$f(y)=\theta y^{\theta-1}$$ and check whether this is biased. I know this is a $Beta(\theta,1)$ distribution and it looks like I only ...
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1answer
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Probability question regarding expectation (Sheldon Ross' book)

One of the numbers 1 through 10 is randomly chosen. You are to try to guess the number chosen by asking questions with "yes-no" answers. Compute the expected number of questions you will need to ...
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Expectation of cubic term for a gaussian random variable.

How do I evaluate the following term? $$\mathbb{E}[x'AxAx'] \qquad \text{for} \quad x \sim \mathcal{N}(\mu,\Sigma) \quad \text{and} \quad A \in \mathbb{R}^{p \times p}$$ By "evaluate" I mean write ...
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1answer
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Expected draws to find $z$ of the same colored balls in $y$ urns

There are $y$ urns Each urn contains $x$ distinctly colored balls. The contents of each urn are identical to start out. You can choose an urn to draw from and draw without replacement, adding that ...
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Independent random variables, sigma

Let $(X_n) $ are independent integerable random variables about the same distribution. Let $S_n=X_1 +.....+X_n$ and $G_n=\sigma(S_n,S_{n+1},.....)$. Calculate : $E(X_1|G_n)$. I tried but i don't have ...
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Expected value-random variables

On $\Omega=[0,1]$ we consider $\sigma$-algebra Borel sets and Lebesgue measure. Let $Y(x)=x(1-x)$. Show that for any integrable random variable $X$: $$E(X\mid Y)(x)=\frac{X(x)+X(1-x)}{2}\quad \text{...
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1answer
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Doubt regarding the maximin value in this question

$A$ and $B$ play the following game: $A$ writes down either number $1$ or number $2$, and $B$ must guess which one. If the number that $A$ has written down is $i$ and $B$ has guessed correctly, $B$ ...
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Expected value ( rectangle) [closed]

We draw 2 numbers $a$ and $b$ from $[0,1]$ . Assuming that $a$ and $b$ are less than $1/2$ what is the expected rectangle field with vertices in points $(0,0) , (a,0), (0,b)$? Please give me some tips
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Least upper bound for the maximum of two random variables

Two independent random variables are not negative and E[X]=a, E[Y]=b. We define a new random variable Z=max{X,Y}. If t>0 and t>max{a,b} calculate the best (least) upper bound for the probability P[Z>t]...
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Determine the expected value and variance.

Suppose you throw a die until you obtain a 6, then throw a coin as many times as you threw the die. What's the expected value of and variance of the number of heads, tails, and heads and tails ...
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1answer
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Expectation of a random variable $X$

9 people are assigned to one of 17 posts uniformly at random and more than one person can be at one post. Letting $X$ be the random variable representing the number of occupied posts, I was tasked ...
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1answer
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Finding $E(X\mid1_{\{X+Y=0\}})$ where $X,Y$ are i.i.d Bernoulli variables

I have two independent random variables $X,Y$ with the same following distributions : $$P(X=0)=P(Y=0)=1-p\,,\, P(X=1)=P(Y=1)=p$$ I want to calculate $E(X\mid 1_{\{X+Y=0\}})$. So Let's define ...
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Does the expectation depends on the time?

I have the interval $[0,1]$ which is divided into $N$ intervals of lenght $\frac{1}{N}$. Each of these $N$ intervals is of the form $[\frac{j}{N}, \frac{j+1}{N}]$ with $j\in \{0, \ldots, N\}$. We call ...
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Points uniformly distributed on a circle

We select randomly two points in the circumference (with length equal to 1) of a circle. Let $X,Y$ be those points (independent and uniformly distributed) and $D$ the arc distance between them. Since ...
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1answer
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Expected value of a formula involving ordered random variables

I have three random real numbers from uniform distribution $[0,1]$, and also a constant $r\in [0,1]$. Supposed the three random numbers are sorted afterwards and denoted $a<b<c$. I want to ...
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1answer
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Half of half by chance

We are given $2^n$ numbers: $1,2,...,2^n$ ($n$ is integer). Then repeat $n$ times the following process: split those numbers in half and flip a coin with $p$ probability for heads. If the result for a ...
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1answer
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The expected weight-ratio between weighted and un-weighted balls when picked from a bin without replacement

The Problem The problem, I believe, can be stated in the following way: Given $K$ white balls all with without weight (one can say that the weight is $0$) and $N - K$ red balls with individual ...
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To find E(X) when the conditional probability is given [closed]

Let Y ~ U(0, 1). The conditional probability density function of X given Y is Y|X   1 , if 0 x y 1, f x | y y 0, otherwise.          Then E(X)
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Expectation, Variance and Moment estimator of Beta Distribution

I'm given a beta distributed random variable: $X \sim \text{Beta}_{(\theta, 1)} =: \mathbb{P}_\theta$. Where $\theta \geq q$ and $$\mathbb{f}_\theta(x) = \theta \cdot x^{\theta-1} \cdot \mathbb{1}_{[...
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1answer
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Find $E[(1-X)^k]$ given $E[X^k] = \frac{1}{(k+1)^2}$

Find $E[(1-X)^k]$ given $E[X^k] = \frac{1}{(k+1)^2}$ The previous part of the question involved finding $E[X^k]$ given $X=AB$, where $A$ and $B$ are uniformly distributed over $[0, 1]$. However, I ...
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Question about Mathematical Expectation [closed]

Balls are taken one by one out of an urn containing 'A' white and 'B' black balls, until the first white ball is drawn. Show that the expectation of the number of black balls preceding the first white ...
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Expectation of function Non-central Chi-square RV

I am trying to calculate the expected value of the function $$f(X) = \gamma \bigl( a,\frac{b}{cX+d} \bigr)~,$$ w.r.t random variable $X$, where $X$ has non-central Chi-square distribution with a ...
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1answer
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variance of $\hat \mu$ from a Gaussian distribution

Assume $X \sim \mathcal{N}(\mu, \sigma^2)$ generates an independent sample $X_1, \ldots, X_n$, where $\sigma$ is known. The exercise is to find an unbiased estimator for $\mu^2$ and its variance. In ...
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1answer
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Compute expected value of the event [closed]

Let $S = \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$ be a sample space with $P(x) = k^{2}x$ where $x$ is a member of $S$, and $k$ is a positive constant. Compute $\mathbb{E}(S)$. Round your answer to the nearest ...
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1answer
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Expectation of a die roll with a coin toss

Let's say we throw a coin and pick a number from [0,1] uniformly. Then we roll a die, and if the coin is heads, we add the number we picked to the number on the die, otherwise we subtract it. What is ...
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1answer
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question on expected value of exponential functions

When is the approximation $\mathbb{E}\left\{{e^{f(x)}}\right\} = e^{\mathbb{E}\left\{f(x)\right\}}$ tight? I mean for which function of f(x) and which conditions? there is any theorem on it?
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Variance of Squared Sum of (weakly) Dependent Random Variables $Var (\sum X_i)^2$

I am trying to work out an example I came up with in which I have to compute the variance of a squared sum of dependent random variables: \begin{align*} W:=\mathbf{V} \left( \sum_{i=1}^n X_i \right)^...
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2answers
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How to express random variable $V(X)$ given function $V(x)$

I have come across a question which reads as follows: For a random variable $X$ with mean $\mu$ and variance $\sigma^2 < \infty$, define the function $V (x) = E((X −x)^2)$ where $'E'$ denotes ...
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calculate expected value given cumulative distribution function

First of all, let me tell you I am not a mathematician, I am trying to understand one problem and I would appreciate if you could help me. I have Gaussian distribution of mean 100 and standard ...