For questions about the expectation of a random variable: computations, upper/below bounds, etc...

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1answer
69 views

I do not understand the last step of this proof. [on hold]

1. PLEASE LOOK THE FOLLOWING PROOF FIRST. 2. Suzu explained the fist several steps to me in this page :Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$ . But I still ...
1
vote
1answer
26 views

If $X\ge 0$ and $a\ge E[X]$, then $P(X\gt a)\ge (E(X)-a)^2/ E(X^2)$ [on hold]

I need help with this problem. Prove that if $X\ge0$ and $E[X^2]<\infty$ then for all $a\neq0$, $E[X]\le a$, we have $$P(X\gt a)\ge\frac {(E(X)-a)^2}{E(X^2)}$$ Progress I have my doubts if ...
0
votes
1answer
19 views

determine how much probability increase with an added condition

Suppose there are $N$ people and $N$ prizes, and only $M$ out of $N$ are valuable. Every time one person is picked randomly, then he pick one prize randomly as well (this prize/person is then removed ...
4
votes
1answer
38 views

Almost surely vs expectation

Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit ...
2
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0answers
25 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
1
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2answers
25 views

Determine the expected value of a geometric distribution given some generic underlying distribution.

This is a variation of the standard waiting time problem. Suppose you have a sequence of variables $$X_0,X_1,X_2,\ldots \overset{iid}{\sim} F(x)$$ where $F(x)$ is continuous. And random variable ...
1
vote
1answer
46 views

Expected value of a Poisson variable conditioned on sum [duplicate]

Setting $$X_1 \overset{d}{\sim} \operatorname{Poisson}(\alpha_1)$$ $$X_2 \overset{d}{\sim} \operatorname{Poisson}(\alpha_2)$$ $$S = X_1 + X_2$$ Find $E[X_1 | S =n]$ My argument is that since $X_1 + ...
0
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1answer
26 views

Compare expectations [closed]

X and Y are two random variables. How would you compare $E[XY]E[XY]$ with $E[X]E[XY^2]$ ? You need to tell which of these is greater/smaller.
2
votes
2answers
49 views

Expected gems needed to level up

There is a game where you have an item and you can raise the level of the item from +0 to a higher value. To do this you have to spend a Gem. First 6 upgrades are 100% successful. The 7th upgrade ...
2
votes
0answers
17 views

Why does Average Log Likelihood

The average log likelihood $$L(W,X) = \frac{1}{N}\sum_{1}^{N} log(p(x_n;W))$$ as defined by the authors in http://www.gatsby.ucl.ac.uk/aistats/fullpapers/217.pdf (first equation, first page, right ...
-1
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0answers
27 views

If $E[X(t)X(s)]=t \land s $. Show that this process has independent increments [closed]

Let $X(t), t\ge0$ be a real-valued Gaussian process with zero mean and the covariance function $\mathbb{E}\left[X(t)X(s)\right] = t \land s $. Show that this process has independent increments.
1
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1answer
19 views

Tower Property and Variance of a Random Variable (Lightbulb problem)

Consider the following question: Type i light bulbs function for a random amount of time having mean (mew)i and standard deviation (sigma)i; where i = 1; 2. A light bulb randomly chosen from a bin of ...
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3answers
54 views

Calculate expected value. [closed]

Can someone give a hint for v). I don't know how to evaluate this integral from 0 to infinity. Thank you!
1
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1answer
37 views

Expected value of variance and variance of variance?

From a certain system dynamical system I am able to calculate the expected error $\mathbb{E}(e_k)$ and the variance $\sigma^2 = \operatorname{var}(e_k) = \mathbb{E}(e_k^2) - \mathbb{E}(e_k)^2$, as ...
1
vote
1answer
35 views

A proof about martingales and variance

We consider a martingale $(S_n)$ with $\mathbb E(S_n^2)<K<\infty$. Suppose that $\mathrm{ Var}(S_n)\rightarrow0$. Prove that $S=\lim_{n\rightarrow \infty}S_n$ exists and is constant a.s. I ...
1
vote
2answers
37 views

Expected Value of Excellent Wine Age

I Ran into this question and I can't find the right way to approach it. We have $n$ different wine bootles numbered $i=1...n$. the first is 1 year old, the second is 2 years old ... the $n$'th bottle ...
1
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1answer
79 views

Secret Santa Probability Question

In a Secret Santa game, a group of N friends each write their name on a slip of paper and place these slips in a bowl. The slips are mixed and each person draws a slip from the bowl. If anyone draws ...
1
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1answer
25 views

Expectation of constraint random walk

Problem description I am currently dealing with a practical problem that can be simplified to something like this: I start by setting a value to 0 Every minute, I try to increase or decrease 1 or ...
0
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0answers
15 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
0
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1answer
26 views

Expectation Of Number Of Dice Rolls

I was asked the following question on a practice test: If two dice are rolled until their sum is seven, or they have been rolled twice, let the random variable ...
0
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1answer
11 views

Identity involving the relation Normal Distribution and Other arbritary Distribution

Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ ...
0
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2answers
46 views

Calculate the mean of the normal distribution function $\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ by integration.

I know that it must be $\mu$ but I cannot get the answer. This is my attempt so far: Normal distribution function = $N(x)=\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ $$\langle ...
1
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1answer
22 views

Central sample moments are asymptotically unbiased

Let $\newcommand\top{\overset p\to}\newcommand\isd{\overset d=}\newcommand\P{\mathcal P}\DeclareMathOperator\var{Var}$$X\isd X_1\isd x_2\isd\ldots\isd X_n$ be independent stochastic variables with ...
1
vote
3answers
62 views

Show that the variance, $\mathbb E((x-\mathbb E(x))^2)$, can be written as $\mathbb E(x^2)-(\mathbb E(x))^2$

This question has been set in the Christmas work for the chemists at oxford uni and the hint that was given in the problem sheet was "does $\mathbb E(x)$ depend on $x$?". There is a derivation on ...
0
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1answer
31 views

What does the expectation value of $x$ mean? Surely it must expectation value of a function of $x$?

How can a value of $x$ have an expectation value? Surely there must be a distribution of values of $x$ for the expectation value to be calculated. Is this the reason for the normal distribution ...
1
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0answers
14 views

Distance between element and its position in permutation

Consider a permutation $\pi$ that is chosen uniformly at random among all permutations of $\{1, \dotsc, n \}$. Let $a_i$ be the position of $i$. We want to find $$E[\sum_{i=1}^n |a_i -i |]= ...
0
votes
1answer
29 views

Expectation of random varible with normal distribution composed with exponential [duplicate]

I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} ...
0
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1answer
28 views

Expected value of the negative portion of sum of poisson random variables

Setting: Defn: for every $x \in \mathbb{R}$ define its negative part by $x^{-} = -x$ if $x \leq 0$, and $x^{-} = 0$ if $x > 0$ Let $\{X_j, j \ge 1\}$, $X_j \overset{d}{\sim} Poisson(1) = \Pr\{X = ...
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0answers
18 views

iid random variables and stopping time

This is Exercise 14.30 from Probability for Statistics and Machine Learning. Let $X_i$ be iid with $E|X| < \infty$, and let $T$ be a stopping time adapted to $\{ X_i \}$. Let $S_n = ...
0
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1answer
83 views

Probability of one candidate passing the test.

There is a class of many students and viva is going on for them. For each student the time taken for viva is given and the probability that he will pass is given.If one student fails the examiner ...
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0answers
22 views

Finding an expectation

I have $n$ dots in a circle and $n$ edges that connect these dots. Each dot is painted red or blue in a probability of $0.5$ each. An edge is blue/red if both the dots it connects are blue/red. Let ...
2
votes
1answer
51 views

3-dim Brownian motion, harmonic function and its expectation

Given $f(x)=\frac{1}{|x+z|}$, a function from $\mathbb{R}^3\backslash \{z\}$ to $\mathbb{R}$, $z \in \mathbb{R}^3\backslash \{0\}$ and $B$ a 3-dim Brownian motion. I had succes showing that this ...
2
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0answers
44 views

Some properties of a random variable

I have absolutely no idea how to show this: Let $X$ be a random variable whose distribution is not degenerate. By considering the function $F( \theta) = \mathbb{E} U( \theta X)$, $\theta \in ...
0
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1answer
32 views

If the expected value is on the boundary of the range, then the random variable is a.s. constant

Let $X$ be a real-valued random variable on $\Omega$, $I\subseteq\mathbb{R}$ be an interval, $X(\Omega)\subseteq I$ and $E[|X|]<\infty$. Why does $E[X]\in\partial I$ imply that $X=E[X]$ almost ...
0
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0answers
22 views

Expectation of geometric mean reversion process

The second part to a question I asked here in which I had to show that the solution to $dX_t = \kappa\left(\alpha-\ln X_t \right)X_t dt + \sigma X_t dB_t$ was $ X_t = \exp \left( \mathbb{e}^{-k ...
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0answers
16 views

Is the set of product distributions with second moment constraint convex?

Consider the following set where $F(X_1,X_2)$ is distribution function of random pair $(X_1,X_2)$: \begin{align*} S=\left\{F(X_1,X_2)\Big| F(X_1,X_2)=F(X_1)F(X_2) \text{ and } E[X_1^2] \le 1, \ ...
0
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0answers
21 views

Expectation and Variance

For a random variable X, suppose that $E[X]=4$ and $Var(X)=9$. Then (a.) $E[(3+X)^2]= ?$ (b.) $Var(2+4X)= ?$ I know the answer for (b.) is $4^2Var(X)=16*9=144$ But I do not know how to do (a.) ...
2
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3answers
38 views

Expectation of the function of a random variable

If a random variable $X$ has finite expectation, is the expectation of the function of $X$, e.g. $$f(X)=\exp(X)$$ also finite? How to prove or disprove?
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0answers
17 views

Help in understanding derivation of density function and expectation maximization

I am unable to understand how the density function is derived in this paper Semiblind System Identification with Symbolic Chaotic Sequences The Authors have ...
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2answers
94 views

Expected value problem: a couple stops having children as soon as they have a child that has the same gender as their first

Givens: $p$ is a real number with $0 < p < 1$ Child is a boy with probability $p$ Child is a girl with probability $1-p$ Anna and Ben stop having children as soon as they have a child that has ...
2
votes
1answer
33 views

Finite limit involving characteristic function implies values of first and second moments

If $$\lim_{c \to 0} \frac{\phi_X(c) - 1}{c^2} = -\frac{\sigma^2}{2} < \infty$$ where $\phi_X(c)$ is the characteristic function of the random variable $X$, then $$E[X] = 0,\qquad E[X^2] = ...
0
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2answers
34 views

Why my solution is not true?

We know that $X_1,X_2$ and $X_3$ are three independent exponential random variables, with rates $\lambda_1,\lambda_2$, $\lambda_3$ respectively. We want to calculate $$E[\max{X_i}| ...
2
votes
1answer
173 views

Probability question involving simulations of picking balls from a bag

I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...
0
votes
1answer
15 views

Clarifications for linearity of expectation

For linearity of expectation to work, do the random variables have to be from the same experiment? S there are two random variables X and Y on the same sample space, which assigns real values $X(s)$ ...
4
votes
0answers
44 views

Crow probability question

Twenty crows land randomly on a wire. Each crow is crowing at the nearest crow. What is the expected number of crows that are not crowed at? I truly have no idea how to approach this problem. I was ...
1
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0answers
38 views

X and Y has the same distribution, prove E(X/Y)>= 1

Let X and Y be two positive random variables with identical distribution ( not necessarily independent), prove $E(X/Y) \ge 1$.
0
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0answers
12 views

How can I find the nonlinear and linear MS estimates of y in terms of x and the resulting MS errors?

If $y=x^3$, find the nonlinear and linear MS estimates of $y$ in terms of $x$ and the resulting MS errors? This is what I got for the nonlinear MS estimation: Since $e=E\{[y-C(x)]^2\}$, $C(x)=x^3$ and ...
2
votes
2answers
77 views

Expectation Random Variables

Say $X$ to be uniformly distributed from $[0,1]$. Say $k_1$ and $k_2$ to be two non negative constants (that is, they take values from $[0,+inft]$. I want to compute the expectation of the following ...
1
vote
0answers
22 views

Inverse Markov inequality with additional information?

Let $X$ be a random variable with values in some finite subset of $\mathbb{N}$ containing 0. Let $a>0$ and $0<\epsilon<1$ be such that $\mathbb{P}(|X-\mathbb{E}[X]|\geq a)\leq2\epsilon\leq ...
0
votes
1answer
16 views

Expected winnings from a game

A game of lucky dip consists of drawing 2 chips without replacement from a box containing 1 green chip, 2 blue chips and 40 white chips. You are paid according to the type of chips drawn. The green ...