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Questions tagged [expected-value]

Questions about the expected value of a random variable.

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Where did I go wrong in proving $\mathbb E[X^{2n}] = \prod_{1 \leq k \leq 2n, k \operatorname{odd}}k$

Let $X$ ~ $\mathcal{N}(0,1)$ Show that: $\mathbb E[X^{2n}] = \prod_{1 \leq k \leq 2n, k \operatorname{odd}}k$ Idea: $\mathbb E[X^{2n}]=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}x^{2n}e^{-\frac{x^2}{2}}...
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Forex rate: Expected value paradox

Let us suppose at present 1 dollar = 1 euro After 1 year There is 50% chance that 1 dollar = .80 euro ...[1] And there is 50 % chance that 1 dollar = 1.25 euro ...[2] Therefore expected value ...
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Is absolute value of the empirical mean unbiased/biased?

Given a random variable $x$ defined on $\mathcal{X}$ and corresponding i.i.d. observations $X_i$, $i=1,...,n$, could we estimate a following probability? $$P\left(\left| \left|\mathbb{E}(x)\right|-\...
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Understanding better conditional expectation : example with tossing coin.

We toss a fair coin three times. So the probability space is given by $(\Omega ^3, \mathcal F^{\otimes 3},\mathbb P^{\otimes 3})$ where $\Omega =\{H,T\}$, $\mathcal F=2^\Omega $ and $\mathbb P\{\omega ...
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Throw a dice-expected value.

We throw the dice three times. Let $X_i$- number of throws, in which we get number $i$. Find expected value and variance random variable $Y=\sum_{i=1}^{6} (-1)^iX_i$. We know, that $X_i=0,1,2,3$. $E(Y)...
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Find $P(Y(x)\gt E[Y(x)])$, where $Y(x)= \min\{i:X_i\gt x\}$ for $(X_i)$ i.i.d.

Let $X_i$ be i.i.d random variables with common cdf F. For a given constant x, define $Y(x)= \min\{i:X_i\gt x\}$. Find $P(Y(x)\gt E[Y(x)])$. What is the limit as $x\to\infty$? My answer: $P(Y(x)=1)=P(...
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Candy Crush - Expected extra moves

Candy Crush have recently introduced a new "feature" where by watching an ad you can get to spin a wheel to get extra moves (you used to have to "pay" for it). This option pops up if you fail to ...
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Expected value of sum from sample without replacement

Suppose we have n numbers, their values are 1 through n, and we sample k times without replacement from these n numbers, the probability of selecting any number is the same, with k < n, let X be ...
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1answer
69 views

Conditional expectation with a third random variable

In this post some basic steps were given as understood, and I've been trying to fill in the gaps without much success. Specifically, the problem calls for random variables $X,$ $Y,$ and $U,$ linked by ...
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Expected Number of Steps to Reach a Destination

Assume there is a row of $k$ tiles. A creature (monkey in some situations, ant in others, frog in others) lies on tile $a$. There is a 50% probability that the creature jumps to tile $a-1$ and a 50% ...
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tighter bound for the expectation of max of sets of functions

Let $X\in\mathbb{R}^{n\times d}$ be a random matrix, and $\{f_k\}_{k=1}^K, m_k \in \mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the ...
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3answers
74 views

Let $T$ be the number of tosses required until three consecutive heads appear for the first time. Find $\textbf{E}(T)$.

A fair coin is tossed repeatedly. Let $A_{n}$ be the event that three heads have appeared in consecutive tosses for the first time on the $n$-th toss. Let $T$ be the number of tosses required until ...
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Is there an upper bound for the mean of a ratio between two random variables?

Is there an upper bound for the mean of a ratio between two random variables ?
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Prove the Martingale Property for a sequence of Random Variables

Let $\Omega = [0,1]$, $\mathcal{F} = \mathcal{B}(0,1)$, P=Lebesgue measure and $X_{n}$ is a random variable defined as $X_n(w)= \begin{cases} 0 \quad \frac{1}{n} < w \leq 1 \\ n-n^2w \quad 0 \leq w ...
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Let X be a Poisson random variable, show that $E[X^n] = \lambda E[(X+1)^{n-1}]$

I am struggling to understand some of the steps of the proof of this problem: Let X be a Poisson random variable, show that $E[X^n] = \lambda E[(X+1)^{n-1}]$. The proof: $$ \begin{align*} E(X^n) &...
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Let $E[X]$ be the expected value of X, what is the meaning of $E[X^n]$

I am interested to understand what's the meaning of using powers in expected value. Both mathematically and maybe intuitively, what does it even mean "the power of a random variable"? Thanks!
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Getting expectation of Gamma functions/Beta functions given a constant

We know $E(X) = \frac{a}{\lambda}$ for gamma function so if to get the expectation for $100x^2(e^{-3x})$ for $x > 0$. This is $X \sim \text{Gamma}(4, 3)$ $$E(X) = 100\int_{0}^{\infty}x^3\left(e^{-...
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Expectation of Reciprocal of Sum of i.i.ds

$X_i$s are real continous i.i.ds. I am trying to prove the conjecture $$ E[\frac{1}{\sum_{i=1}^nX_i}] = E[\frac{1}{nX_i}] $$ I can't seem to be getting anywhere. Prove or disprove.
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Expected number of well addressed parcels

Consider the following mad postman scenario. The mad postman has $n$ parcels which should go to $n$ different destinations. However, the mad postman assigns destinations to the parcels randomly. What ...
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1answer
56 views

Candy machines and optimal strategy in terms of expected value

Problem We have three candy machines: call them G (good), B (bad) and M (mixed) . G always gives you a candy when you put 1\$. B never gives you a candy when you put 1\$. M gives you a candy with ...
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44 views

Expectation of the supremum of a sequence of random variables

Let $\Omega = [0,1]$, $\mathcal{F} = \mathcal{B}(0,1)$, P=Lebesgue measure. Let $X_n(w)= \begin{cases} 0 \quad \frac{1}{n} < w \leq 1 \\ n-n^2w \quad 0 \leq w \leq \frac{1}{n} \end{cases}$ The ...
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21 views

bound the expectation of max of sets of functions

Let $X\in\mathbb{R}^{n\times d}$ be a random matrix, and $\{m_k\}_{k=1}^K, m_k \in \mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the ...
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0answers
17 views

Expected Cost in the Second Price Auction

I am going through this paper https://arxiv.org/pdf/1803.02194.pdf on Bidding Machine. In the section 3.1 Problem definition, I got the part about finding the probability of winning at bidding price $...
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Proving the Linearity of Expectation from the opposite side

An alternative method of variance is $$E[X^2]-(E[X])^2$$ and proving that $E[(X-E[X])^2] = E[X^2]-(E[X])^2$ is done. But how about the other way around $E[X^2]-(E[X])^2 = E[(X-E[X])^2]$, how can you ...
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Numerical approximation to Beta moment generating function

I have a Beta random variable $X \sim \text{Beta}(\alpha, \beta)$, and I'm interested in $\mathbb{E}[e^{2X}]$. The Beta distribution moment generating function is $$f(t) = {\displaystyle 1+\sum_{k=1}...
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1answer
28 views

Finding the probability of a random variable given its expectation

I'm taking the Algorithms: Design and Analysis II class, one of the questions asks: Suppose $ X $ is a random variable that has expected value $ 1 $. a) What is the probability that $ X $ is $ ...
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1answer
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Conditional expectation $\mathbb{E}[\mathbb{E}[X\vert Y,Z]\vert Z]$

Problem When proving one result in the statistical learning theory course, the instructor uses $$ \mathbb{E}[\mathbb{E}[X\vert Y,Z]\vert Z]=\mathbb{E}[X\vert Z] $$ but I am not sure why this is true. ...
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Variance decomposition of Poisson process

Let $N(t)$ a poisson process with intensity $\lambda$ and $X$ a positive random variable independent of $N$. Let $f$ a real valued increasing function with $f(0)=0$. Consider $C(t):=N(Xf(t))$. ...
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28 views

Let $ X \sim \operatorname{Exp}(\lambda =1), Y\sim U(1,2) $ be independent continuous variables. What is $E(\frac{x}{y})$?

I'm struggling to understand how to start the following: Let $ X \sim \operatorname{Exp}(\lambda =1), Y\sim U(1,2) $ be independent continuous variables. What is $E(\frac{x}{y})$? Thanks :)
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1answer
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Notation and absolute continuity for probability measures

A Question on Notation: I've realized that different notations are used in probability theory when evaluating an integral, and I am unsure as to how they "work" together, whether they're completely ...
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explanation of E[X] = Sup(E[Y] : Y a simple r.v.)

Could someone explain me the meaning of the following expected value of a positive random variables $X$? $\mathbb E[X] = \sup(\{\mathbb E[Y] : Y\text{ a simple r.v. with }0 < Y < X\})$ where ...
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1answer
22 views

Calculation of variance of complicated random variable ( white noise discretization )

so I have been doing some state estimation and in one part of my work it is necessary to discretize a continous time differential equation with white noise. I understood the discretization process for ...
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20 views

Expected value of Bernoulli variable (for Linear regression model)

Can anybody explain how the following reduction happens:
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1answer
19 views

What is the expectation of the product of a normalized (complex) gaussian vector and its hermitian transpose?

I have a complex Gaussian vector $\mathbf{x} \triangleq (x_1; x_2; ...; x_n) \in \mathbb{C}^{n \times 1}$, where $x_i \sim \mathbb{CN}(0,1)$ i.i.d., and $\mathbf{y} \triangleq (y_1;y_2;..;y_n)\in \...
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2answers
38 views

Prove that if $\mathbb{E}[g(Y)]\leq \mathbb{E}[g(X)]$ for every nondecreasing function $g$ then $F_X\le F_Y$ [closed]

Let $X,Y$ random variables. Every coninuous and not-decreasing monotone function $g:\mathbb{R}\to[0,1]$ fulfills $\mathbb{E}[g(Y)]\leq \mathbb{E}[g(X)]$. Prove that $F_X(t)\leq F_Y(t)$. I don't have ...
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Understanding equality regrading expectation of random matrices

I'm reading the following article on Latent Tree Structures (I added a link at the end of the post) : "Spectral Methods for Learning Multivariate Latent Tree Structure". I'm trying to understand the ...
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1answer
15 views

Find expectation of of max(X,5) from given probability mass function

Let $X$ be random variable with pmf $P(X=n)=\dfrac{1}{10},n=1,2...10$ $0$ otherwise Find $E(max(X,5))$ Let $Y=max(X,5)$ $P(Y=y) = \begin{cases} 5, & \text{if $X \le5$ } \\[2ex] X, &...
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Convergence in probability towards a constant

Prove that if a sequence $X_1,X_2..$ of random variables satisfies the following conditions: $$1.lim_{n\to\infty}\mathbb{E}(X_n)=a,\space a\in\mathbb{R}$$ $$2.lim_{n\to\infty}\mathbb{V}(X_n)=0$$ Then ...
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The expected value of a random function

Suppose we have a 'random function' $f(x)$ that will be one of $n$ functions $f_1(x), ..., f_n(x)$ with probabilities $p_1, ..., p_n$. Intuitively, it might seem sensible to define $$E[f(x)] = \sum_{...
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1answer
39 views

Let $X_1,X_2,X_3$ be iid. U($0,1$) random variables. Then what will be the value of $E(\frac{X_1+X_2}{X_1+X_2+X_3}$)? [closed]

Let $X_1,X_2,X_3$ be iid. U($0,1$) random variables. Then what will be the value of $E(\frac{X_1+X_2}{X_1+X_2+X_3}$) ?
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Expectation of $A^TXA$ for random $A$ and $X$

Suppose there are two random matrices (distribution unknown), denoted as $A$ and $X$, both in the $\mathbb{R}^{n \times n}$ space. It is known that $\|A\| \leq 1$ (for any $p$-norm) and $E[X]\geq0$. I ...
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1answer
56 views

Expected coin tosses until at least h heads and t tails are obtained

I am also wondering about the same question for any Bernoulli variable. I have got as far as saying for any Bernoulli variable: $$ \mathbb{E}[X] = h \sum\limits_{n \geq h + t}^{\infty} {n \choose h} p^...
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Compute $E(g)$, where $g=2-|f|$ if $|f|\le\phi$ and $g=0$ otherwise, and $\phi$ is uniformly distributed over $[0,1]$

$$E\left[\begin{cases} 2-|f|,& |f|\le\phi\\\\ 0,& \text{else} \end{cases}\right]$$ where $\phi$ is uniform distribution over $[0,1]$ I find it difficult to calculate it using the integral. ...
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1answer
39 views

What is the expected value from these two different coin tossing games?

Consider these two games: Game $1$: Toss $4$ coins. If coins $1$ and $2$ are heads, you win $\$5$. If coins $3$ and $4$ are heads, you win an additional $\$5$. Game $2$: Toss $3$ coins. If coins $...
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Compute $\mathbb{E}\big[\exp(XY+X)\big]$ where $X, Y$ are independent uniformly distributed over $[0,1]$ r.v's.

Compute $\mathbb{E}\big[\exp(XY+X)\big]$ where $X, Y$ are independent uniformly distributed over $[0,1]$ r.v's. I first computed $\mathbb{E}\big[\exp(XY+X)\mid X\big]$. Because $X$ and $Y$ are ...
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1answer
35 views

Expected Value Iteratively picking divisors of number [closed]

You have a positive integer $n$ written on a blackboard, and you perform $k$ iterations of the following procedure: say the current number is $v$. Pick one of the divisors of $v$ (possibly 1 and $v$...
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3answers
57 views

Expected number of consecutive guesses to get a given sequence of numbers

I have a lock on my dorm door that's really stupid. Basically, it just checks whether or not the sequence of numbers I've put in is the combo, whether or not the lock was reset between guesses. So let'...
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1answer
45 views

How do I apply Law of Large Numbers in this question

Let $K_{0}=1$, and in every round $i =1,...,n$ a player puts all capital $K_{i-1}$ in. A fair coin is thrown. If it falls on Heads he receives $\frac{5}{3}$ of his capital outlay, if it falls on Tails ...
3
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1answer
36 views

Bounding expectation of stopping time

Let $(X_{t})_{t\ge0}$ be adapted to $(\mathcal{F}_{t})_{t\ge0}$ with continuous trajectories. Assume that $X_{0} = 0$ and $X_{t}^{4} - 3t^{2}$ is a martingale with respect to $(\mathcal{F}_{t})$. Let ...
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0answers
24 views

Obtaining derivatives for the moments of random variables

I am currently working through a problem and am having trouble interpreting my results. I start with a system whose dynamics are described by the equation: $$ \frac{dC}{dt} = f(C(t),\theta) $$ ...