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

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How to find expected number of games between two players?

I couldn't understand an answer to this question, so I'm asking it again. Can someone explain the answer or solve it by another method? The one think I didn't understood in answer is why ...
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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 ...
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1answer
23 views

Markov Chain Expected value

Let $(X_n)_{n\in\mathbb{N}}$ be a Markov chain with State space $E=\{1,2,3\}$ and transission matrix $$P=\begin{bmatrix} 0 & 1/3 & 2/3 \\ 1/4 & 3/4 & 0 \\ 2/5 & 0 & ...
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How many draws to collect all items? [duplicate]

Let's say there's a box with n different items in it and the goal is to collect all different items. This is done by drawing a random item with probability of 1/n, writing the outcome and putting the ...
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Expectation Value of sample average [on hold]

A sample consists of 1 to n numbers with mean = a. What is the expected value of the sample average. And what is the physical significance of this value.
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A composition of random variables, finding Expectation and Variance??? [on hold]

The number of defects per yard $Y$ for a certain fabric is known to have a Poisson distribution with parameter $\lambda$. However, $\lambda$ itself is a random variable with probability density ...
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32 views

$E(\bar{Y})=\bar{Y}$? [Linear Regression]

A book, I am reading, derives the covariance between $b_0$ and $b_1$ as follows: By definition, $$Cov(b_0,b_1)=E[(b_0-Eb_0)(b_1-Eb_1)]$$ $$=E[(b_0-\beta_0)(b_1-\beta_1)]$$ ...
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1answer
23 views

Expected score in marksmanship competition.

Problem: Marksmanship competition at a certain level requires each contestant to take ten shots with each of two different handguns. Final scores are computed by taking a weighted average of 4 times ...
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1answer
30 views

Expectation of the derivative of a random process

Let's have a Random Process $Y(t) = X(t) + 0.3 X'(t)$ Mean of $X(t) = 5t$ Question : Find the mean function of $Y(t)$ What I did : $E(Y) = E(X) + 0.3\cdot E(X')$ ? I don't know if I have ...
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2answers
31 views

Translating expected values between two sets of related iid variables

The setting: $\mu$ is a probability measure on $\mathbb{R}$, $f: \mathbb{R} \to [0, \infty)$ so that $0 < ||f||_{L^1(\mu)} < \infty$, and $v$ is another probability measure defined by $v(A) = ...
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1answer
44 views

Proof and interpretation of $\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]$

First, I understand that $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$, but how to prove that $$\mathbb{E}[\mathbb{E}[X \mid Y, Z] \mid Z] = \mathbb{E}[X \mid Z]?$$ Second, for ...
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1answer
39 views

show that $\mathsf E(Y|X)=\mu_2+\rho\dfrac{\sigma_2}{\sigma_1}(X-\mu_1)$

Suppose $X$ and $Y$ have a joint distribution with finite means and variances respectively given by $\mu_1,\mu_2,\sigma_1^2,\sigma_2^2$ with correlation coefficient given by $\rho$. Further suppose ...
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0answers
26 views

What is $E[\cos X]$ where $X$ is lognormal?

I was asked in an interview to compute $E[\cos X]$ where $X$ is lognormal. I tried using lognormal's characteristic function (Taylor series representation, which is divergent) and $\cos ...
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10 views

Roots of Expectation of random variable raised to powers [closed]

Given 1$\le$q$\lt$p And let X be a random variable such that $E(X)^p\lt\infty$
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31 views

Does a joint PDF depend on whether one variable comes from a subset of the other?

Does $f(I,\omega) = f(I)$ where $f(\bullet)$ is a PDF and $\omega \subset I$? Here $\omega$ and $I$ and both are $\sigma$ algebras, with $\omega \subset \Im$, and $I \subset \Im$. To be clear, the PDF ...
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1answer
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Expected value of a circuit given $f_{X, Y}(x, y) = k(x + y)$

HW problem, not sure where I'm going wrong on this. Find $E(R)$ for a two-resistor circuit similar to the one described in Example 3.9.2, where $f_{X, Y}(x,y) = k(x + y)$, | $10 ≤ x ≤ 20$, $10 ≤ y ≤ ...
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3answers
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Exponential Random Variables and either cases of a Conditional Expectation

We are given a random variable X which has an exponential distribution of parameter λ=1. $$X\sim\exp(λ=1)$$ We know that $$E[X]=\frac{1}{λ}$$ Hence for us $E[X]=1$. By virtue of the memoryless ...
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2answers
36 views

Expectation maximum between a constant and a random variable

Let $X$ be a random variable. For sake of simplicity assume it is uniformly distributed from $[0,1]$. Let $c$ be a constant in the same interval. How do I express $E[\max(X,c)]$ in such a case?
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20 views

Expectation Maximization, E step clarification

I'm a bit confused about the E step on the EM derivation for a mixture of multinomials. I don't think I'm all the way there, but I'm confused about how to proceed. Is the E step, solving for tau? Or ...
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1answer
35 views

$E[1_{\lbrace P_T-P_{\tau_n}=0\rbrace}\int_{\tau_n}^T h(s)dN_s]=0?$

If $P_t$ is a standard Poisson process, and $N_t=P_t-t$ the associated martingale then $\int_0^t h(s)dN_s$ is a martingale (assuming that h satisfies the neccessary hypothesis). Thus, considering ...
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0answers
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Closed-form expectation of CES function of a random variable?

I am faced with the following function, called CES (constant elasticity of substitution), of the continuously-distributed random variable $\epsilon$: $f(\epsilon) = (a^\sigma + ...
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1answer
30 views

Fisher information of the Rayleigh distribution

Problem description: Find the Fisher information of the Rayleigh distribution. I was satisfied with my solution until I saw that it disagreed with the solution obtained in one of the problem sets from ...
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17 views

equality involving the arrival times of a poisson process

Let $P_t$ be a Poisson process with arrival times $\tau_1,\tau_2,\dots$ and $h$ a bounded function, $F$ a square integrable function of the arrival times of $P_t$ until the time T . I am wondering if ...
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1answer
49 views

Expected number of pattern in n coin-tossing without overlapping

$N$ is the total number of tossing a fair coin. What is the expected number of occurrence HHH in this $N$ tosses? When overlapping is allowed, e.g., HHHH counts as 2 occurrence, then it is easy to ...
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58 views

Convergence of Expectations (cont'd)

The question is related to this question. Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ ...
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3answers
26 views

Expected value of absolute value of trades

Suppose each day for $365$ days, we flip a coin. If it lands heads, I get \$10. If it lands tails, I lose \$10. What is the expected value of the absolute value of the amount of money I have (or owe) ...
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1answer
50 views

coin toss and conditional expectation

Very basic question, yet I'm confused since my results are different from what I would expect. Coin toss exercise: let $X_n = sum_{n}{B_i}$ is a random variable where $B_i=1$ if it is head and -1 if ...
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1answer
141 views

$n$ balls are thrown randomly into $k$ bins - how many are empty?

A large number of variants of this question were already asked here, including these - one, two, which are close, but none seem to answer my question. Assume that $n$ balls are thrown randomly and ...
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1answer
29 views

Correlation coefficient and Expectation of two dimensional normal distribution.

Random variable (X,Y) is normally distributed. Conditional expectations are $E(X|Y=y)=0.25y + 2$ $E(Y|X=x)=x-2$ How can i determine correlation coefficient and when that is known, the expectations ...
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1answer
31 views

Expected winnings of a slot machine

Of the three wheels on a slot machine where each wheel has ten items: First wheel has 2 flowers, 4 dogs, 4 houses. Second wheel has 6 flowers, 3 dogs, 1 house. Third wheel has 2 flowers, 3 dogs, 5 ...
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2answers
43 views

Expected time to get x units away when only able to move 1 unit either way

I know this is a common problem, but this problem has been bugging me after someone asked me it, and I can't find the answer anywhere on the Internet. Say we have a number line, and we start at the ...
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2answers
38 views

Convergence of Expectations

Suppose $\{X_n\}$ is a sequence of non-negative random variables such that $$EX_n<\infty, \text{ }\lim_{n\rightarrow \infty}EX_n =\infty$$ and $\lim_{n\rightarrow \infty}X_n$ exists a.s. May I ...
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2answers
20 views

Showing 1/E(W) <= E(1/W)

How do I show that $\displaystyle \frac{1}{E(W)} \leq E\left(\frac{1}{W}\right)$ for a positive random variable W? I may be intended to use the Cauchy-Schwarz Inequality, $[E(XY)]^2 \leq ...
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2answers
28 views

Proving the expected value of the square root of X is less than the square root of the expected value of X

How do I show that $E(\sqrt{X}) \leq \sqrt{E(X)}$ for a positive random variable $X$? I may be intended to use the Cauchy-Schwarz Inequality, $[E(XY)]^2 \leq E(X^2)E(Y^2)$, but I'm not sure how.
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Integrations of functions with different inputs

I have the following expression \begin{equation} \int f(x,y)g(y) dy = 0, \quad \forall y \in \mathbb{R} \end{equation} and $g(y)\geq 0$. Are there any conditions on the function $f(x,y)$ that we can ...
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2answers
49 views

Mean of a Cauchy Distribution

Why is the mean of a Cauchy distribution undefined? Surely, it should be $0$ by symmetry? $$\int_{-\infty}^{\infty} {\frac{x}{\pi (1+x^2)}} dx =0?$$
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Find expected value of root of a sample variance

It is easy to compute $E(s^2)=E(\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2)=\sigma^2$. My question is - how can E(s) be calculated? I'm trying to use $\frac{s^2(n-1)}{\sigma^2}$ has Chi-square ...
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2answers
50 views

Expected number of files to find all viruses

So there are 12 files in total and 3 of them contain viruses. If a file with a virus is selected, it is removed and a new file is then selected. What is the expected number of files that need to be ...
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probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] ...
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How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
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1answer
77 views

What does “taking expectation w.r.t some random variable” mean in this probability calculation?

I am trying to calculate the following probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where, $$A_i \sim \exp(\lambda), \quad S_i \sim ...
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1answer
43 views

Determine Random Variable

We have a random variable $X$. Given the values for $E(X), E(X^2), E(X^3), ...$, is it possible to determine the distribution of the random variable X? PS: Here $E(X)$ is the expected value of the ...
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1answer
119 views

A Liar's Paradox Mathematically

The premise (you can skip to the mathematical part below): You are driving back to the town where you were born. You haven't been home for a very long time and you are unsure if you are even on the ...
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0answers
50 views

Expectation of geometric random variable

Let $X$ be geometric random variable with parameter $p$. How to prove that: (1) $E[X-1|X>1] = E[X]$ (2) $E[X^2|X>1] = E[(X+1)^2]$ Author explains the fact below and states it is used ...
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1answer
60 views

Evaluating Expectation of stochastic process

Say, for $u>t$ we have a stochastic process given by : $$ r_u=r_t + \int_t^u\theta_s ds+\sigma\int_t^udW_s, $$ where $W_t$ is a brownian motion, $\sigma$ is a constant and $\theta_t$ is some ...
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1answer
25 views

Calculating Expectation from CDF

The CDF is defined as follows: $$ F(x) = \begin{cases} 0,\qquad x \lt 0 \\[3ex] \frac{x^2}{18}+\frac{x}{6}, \quad 0 \le x \lt 3\\[2ex] 1,\qquad x \ge 3 \end{cases} $$ And i have to calculate the ...
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Closed form for the expectation of random variables built from an ordered sample

Let $n, k \in \mathbb{N}^*$ with $2k \leq n$. Let $Y_1 < \ldots < Y_{2k}$ be an ordered sample of size $2k$ in $\{1, \ldots, n\}$, and let $B_1, \ldots, B_k$ be i.i.d. Bernoulli random ...
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1answer
12 views

What is the expectation of the product of dependent, normal random variables

Question: Let's say I have $X \sim N(\mu_1, \sigma) $ and $Y \sim N(\mu_2, \sigma) $. I know that $ cor(X,Y) = \rho $. What is $E(XY)$? What I've tried Based on a similar question where X and Y are ...
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10 views

An expectation of white balls and black balls

Let us consider an urn where there are $b$ black balls and $w$ white balls. Now in each stage $r$ black balls are put in the urn and then $r$ balls are removed at random from the total $b+r+w$ balls. ...
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1answer
26 views

Expectation of continuous uniform distribution

I'm having a problem with a basic probability problem. There is a stick which is 4 units in length, we break it in two pieces and the breaking point is randomly distributed. After this we form a ...