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

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The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
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1answer
12 views

coincidence of recurrent random processes with infinite expected periods

That subject might not be quite accurate, but let me clarify. At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, ...
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2answers
31 views

Expectation and Variance of random variable [on hold]

I roll a fair die four times. Let X be the number of different outcomes that I saw. (For example, if the die rolls are 5, 3, 6, 6, then X = 3 because the different outcomes are 5, 3, and 6). Find the ...
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3answers
21 views

Expectation of a gamblers winnings

Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after ...
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16 views

Matlab - calculate variance and expected value

I have a matlab question to solve some probability theory, expected value and variance. the code so far: ...
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42 views

Alternating series of compositions of triangular numbers

I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls. Each time I add a ball to $S$, it may dislodge another ball with probability proportional to ...
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1answer
39 views

Expected value of $X^{2n}$ where $X \sim N(0,1)$ [on hold]

The question is: Show that if $X ∼ N(0, 1)$ has the standard normal distribution then $E[X^{2n}] = \frac{2n!}{2^{n}n!}$ Hint: compute the integral $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} ...
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$P(X ≥ a) ≤ \frac{Var(X)}{Var(X) + a^2}$ when E(X) = 0

The full question is: Show that if $$E(X) = 0$$ then $$P(x\geq a)\leq \frac{Var[X]}{Var[X] + a^2}$$ Also show that there is an X for which the equality holds. I was able to note that: $$Var[X] = ...
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Condicional Expectation when $\mathbb{E}[X] = \infty$.

Let $(\Omega, \textit{F}_0, \mathbb{P})$ and $\textit{F} \subset \textit{F}_0$. Suppose $X \geq 0$ and $\mathbb{E}[X] = \infty$. Then there is a unique $Y \textit{F}$-measurable with $0 \leq Y \leq ...
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35 views

How to find approximate probability of obtaining population variance between $10$ and $15$? [on hold]

A sample of $15$ observations is taken from a normal population. It has been calculated that the sample mean is 30 and the sample variance is $12.1$. Find the approximate probability of obtaining ...
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Expectation related to Wiener process using strong Markov property

Can you help me to understand a result I found in Krylov's book "Introduction to stochastic calculus". First, I will introduce some notations: $w_t,t\ge 0$ denotes a Wiener process. ...
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38 views

Expected Value of exponetial form

calculate expected value of $y$ when $$ f(y)=\frac{e^{\frac{y(1-\theta)}{\theta}}}{\theta} $$ I think $E(y)= \theta$ can I verify it? Thanks a lot!
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2answers
17 views

Show $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$

Suppose X and Y are integrable random variables on the measure space $(\Omega,\mathcal F, P)$. Im trying to show that $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$ but I got ...
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I found $E(Σ_{j=0}^{k-1}η_j-Σ_{j=0}^{k-1}E(η_j|G_j))^2=Σ_{j=0}^{k-1}(E(η_j)^2-E(E(η_j|G_j)^2)$ in a book with faulty assumptions on the objects

In Stochastic Equations in Infinite Dimensions (Second Edition) on page 109, the authors state the following: If $\eta_0,\ldots,\eta_{k-1}$ are random variables with finite second moments and ...
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1answer
43 views

Approximately calculate the probability that the average lifetime of all the bulbs in a particular box exceeds $2500$ hours.

I'm trying to solve the following questions Suppose that the lifetimes of light bulbs are independent, exponentially distributed random variables with a mean of $2000$ hours each. 1) Calculate ...
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23 views

Expectation in Bayes rule

Let $L(\theta,\delta)=(\theta-\delta )^2e^{\frac{(\theta-100)^2}{900}}$ with $X\sim N(\theta,100)$ and $\theta\sim N(100,225)$ find the bayes rule. I already founded that posterior is ...
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2answers
23 views

Expectation and variance of matrix valued random variable

Suppose I have a discrete matrix-valued random variable $X$, that is, I have defined a set of fixed matrices $\{Y_i\}_{i=1}^n$, and the random variable $X = Y_i$ with probability $\frac{1}{n}$. Is ...
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2answers
55 views

Quantum mechanics. [closed]

If $P(x) =Axe^{-x^2/a^2}$ for $x > 0$ and $P(x) = 0$ for $x < 0$, find $A$ such that $$\int_{-\infty}^{\infty}P(x)dx=1$$ And hence calculated the expected value of $\left\langle x\right\rangle$ ...
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expectation of stopping time in Wiener process

Let $(W_t)$ be a Wiener process and for $a>0$ define stopping time: $$\tau = \inf \left\{t>0: W_t + at = 5\right\}$$ a) show $\tau < \infty$ a.s; b) compute $\mathbb{E}\tau$. I have done ...
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Help understanding integral proof of variational auto-encoder

I'm trying to understand the key proof of the parametrization trick of this paper "Auto-Encoding Variational Bayes" http://arxiv.org/pdf/1312.6114v10.pdf. It's section 2.4 on page 4. which states ...
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1answer
18 views

What is your expected waiting time if limousine inter-departure times follow an exponential distribution?

Limousines depart from the railway station to the airport from the early morning till late at night. The limousines leave from the railway station with independent inter-departure times that are ...
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1answer
23 views

Expectation of the fraction a random function covers its range

Preamble: The number of onto functions from a set of $m$ elements to a set of $n$ elements is, as stated in this answer, computed as follows: $$n!{m\brace n}\;.$$ Now, let's count the number of ...
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2answers
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expectation of the number of empty cells

You are given a random number, $N$, of balls, where $N$ has a Poisson distribution with parameter $\lambda > 0$. You then place these balls one by one among $r$ ($\geq 2$) cells according to the ...
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Why is the following expectation inequality true?

If $X_1$ and $X_2$ are random variables, why is the following inequality true: $$|\mathbb{E}X_1 - \mathbb{E}X_2| \leq \mathbb{E}|X_1 - X_2|$$
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Expected value of size of randomly choosen subset [closed]

Given a set S such that |S|=n. Let X - numbers of items in randomly choosen (but non-empty) subset of S. Each subset have the same probability to be choosen. Find E(X).
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1answer
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What is E[X]? What is Var(X)?

The number of accidents X that a person has in a given year is a Poisson random variable with mean Y . However, Y ∼ Uniform ([2, 4]). Calculate: (a) E[X] (b) Var(X) Extra My understanding of the ...
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1answer
12 views

expectation and variance about the binomial distribution

enter image description here Suppose we obtain 40 bananas and separate then into sets of “light bananas” (those that weigh less than 4.0 ounces) and “heavy bananas” (those that weigh more than 4.0 ...
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Rate of convergence of $\mathbb{E}\left[1_{Y<h(X)<(1+\gamma)Y}\right]$ as $\gamma$ goes to 0

I am stuck on a probability question I do not manage to solve. Let me introduce some context. We have: -Two random variables, not independent X and Y. -A continuous function h. For $\gamma >0$, ...
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1answer
47 views

Expected value of reaching -7 or 10

Suppose that there is a man standing on the origin of the real line and plays heads or tails. Everytime he gets a head, he moves $1$ unit right and everytime he gets a tail, he moves $1$ unit left. ...
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Rolling a die until two rolls sum to seven

Here's the question: You have a standard six-sided die and you roll it repeatedly, writing down the numbers that come up, and you win when two of your rolled numbers add up to $7$. (You will ...
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1answer
63 views

Assuming the two people wait for each other, what is the expected waiting time?

Two people agree to meet at a restaurant. Assume their arrival times are independent and uniformly distributed on the one hour interval from 1:00–2:00 p.m. Assuming the two people wait for each other, ...
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Tight upper bound for expectation of function of a positive and bounded random variable

This problem popped up in my research. Let $X$ be a positive and bounded ($X\in (0,B) \ a.s.$)random variable with degrees of freedom $d$ and noncentrality parameter $\lambda$. My goal is to find ...
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Expectation of the maximum of random variables

I'm trying to get $E(\max \{ a-X, b-X-Y, 0 \})$, where $X$ ~ $N(0,\sigma^2)$, and $Y$ ~ $N(\mu, \gamma^2)$, and $X,Y$ are independent. I've been trying to figure this out by doing, $E(\max \{ a-X, ...
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Derive the Informaiton Matrix of a Maximum Lilelihood Function

Please click here to view the math reasoning process. Can any one please help to explain, step by step, how we can calculate the final result of the information matrix? Thanks a lot.
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1answer
28 views

Inequality of the expectation vs monotone function

I'm reading understanding machine learning and several of the latest lemmas I've studied involved this inequality which I've searched for but found no justification of whatsoever. Could anyone point ...
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2answers
32 views

Average time between successive occurrences of earthquakes?

In any given year, the probability of an earthquake greater than Magnitude $6$ occurring in the Garhwal Himalayas is $0.04$. The average time between successive occurrences of such earthquakes is ____ ...
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Expectation and variance (unknown problem type)

I'm having trouble identifying what type of problem the following question is: A student is trying a new study strategy for the final exam. There are four topics to study for the exam and each day ...
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0answers
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Gaussian expected inner product with respect to fixed matrix

Suppose $R\in\mathbb{R}^{p\times p}$ is a a fixed matrix (it can be asymmetric, non-positive definite, and so on). Then, I would like to find a formula for the expectation $$\mathbb{E}_{x\sim ...
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1answer
37 views

Relation between $\exp(\mathbb{E}[X])$ and $\mathbb{E}[\exp(X)]$

I have a positive random variable $X$, can I prove any relation between $\exp(\mathbb{E}[X])$ and $\mathbb{E}[\exp(X)]$. To elaborate, I don't have the PDF for $X$ but I do have its mean and from it I ...
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0answers
12 views

Expectation over Wishart distribution of rotated trace

Suppose $S$ is drawn from the Wishart distribution $W_p(V,n)$ with $n$ degrees of freedom and positive definite scale matrix $V\in\mathbb{R}^{p\times p}$. Then, given a matrix ...
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1answer
18 views

identity with Poisson process

I want to show (if it is true) that if $\tau$ is a stopping time and $\mathbb{E}\tau < \infty$ then $\mathbb{E}N_\tau = \mathbb{E}\tau$ for $N_t$ - Poisson process with parameter $1$. I started ...
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1answer
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Mean of binary random variable with probability $.75$ of getting $1$

Given random variable $X$ that takes on $0$ or $1$ with probability of $0.25$ and $0.75$ respectively. What is the mean of $X$? The answer sheet says $0.5\cdot1+0.5\cdot0=0.5$ which I do not ...
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2answers
15 views

Relation between expectation and sample points

Suppose that the expectation of a random variable $X$ is $5$. Which of the following statements is true? There is a sample point at which $X$ has the value $5$. There is a sample point at which $X$ ...
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1answer
31 views

Lower bound for expectation of absolute sum of Rademacher

Let $\epsilon_i$ be i.i.d. Rademacher random variables (i.e., $\epsilon_i$ takes value $\pm 1$ with equal probability). The upper bound $\mathbb{E} |\sum_{i=1}^n \epsilon_i| \le \sqrt{n}$ follows from ...
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1answer
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Find expected time to reach a point on the x-axis from another point on the x-axis

You are standing on a point on the x-axis and want to reach another point on the x-axis. You are allowed to move left or right from your current position and this move is chosen uniformly at random. ...
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1answer
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Expectation of random variable and measure transform theorem

Let $(\Omega,B,\mu)$ be a probability space where $\Omega$ is $[0,1]$, $\mu$ the Lebesgue measure, $B$ the Borel $\sigma$-algebra of $[0,1]$ and $f(w)=1-w$ be a random variable. Let $\phi:\mathbb ...
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The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
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1answer
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($N_t$) is Poisson process with $\lambda = 1$. Calculate $E(N_2|N_1)$ and $E(N_1|N_2)$

($N_t$) is a Poisson Process with constant rate $\lambda = 1$. $1)$ Calculate $E(N_2|N_1)$: So this is how far I've gotten: Let $N_2 = N_1 + (N_2 - N_1)$ $E(N_2|N_1) = E(N_1|N_1) + E(N_2 - ...
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1answer
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Differentiating Laplace transform of random variable

Let $Y$ be a random variable and $A$ an event, such that $g(q):= E(e^{-qY} ; A)$ exists for all $q \geq 0$. (Here $E(X ; A) := \int_{A} X \hspace{3pt} dP$ for a random variable $X$). I want to check ...
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Expectation of random variable $f(w) = w$.

Expectation of random variable $f(w) = 1-w$. Is this right? $Fx(x) = \mu_f$ $ \left\{\begin{matrix} \mu(\emptyset) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ x < 0 \\ ...