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

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23 views

$E[W_s\int_s^tW_sds]$: are $W_s$ and $\int_s^tW_s$ independent?

$E[W_s\int_s^tW_sds]$ Let $W_s$ be a brownian motion, I have to compute $E[W_s\int_s^tW_sds]$. Are they independent?
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1answer
23 views

Calculation of Conditional Expectation

I have problems with the following exercise: Let $\Omega=[-\frac{1}{3},\frac{1}{3}]$, $\mathcal{F}=\mathcal{B}(\Omega)$ the Borel-$\sigma$-algebra on $\Omega$ and P the Lebesgue-measure. ...
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1answer
34 views

$E[W_s\int_s^t W_sds]$, $W_s$ is a brownian motion

Let $W_s$ be a brownian motion, I found $E[W_s\int_s^tW_sds]$ in a much longer exercise but I don't know how to compute it. Any suggestion?
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0answers
30 views

expectation lognormal and normal

I have two random variables $X\sim N(m_{X},\sigma^2_{X})$ and $Y\sim N(m_{Y},\sigma^2_{Y})$ both normally distributed and they're jointly normally distributed as well with correlation $\rho$. I am now ...
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1answer
21 views

Basic question about conditional expectation

Consider $X$ and $Y$ tow random variables $\mathcal F_2$-measurable where $\mathcal F_1$ and $\mathcal F_2$ are two $\sigma$-algebras such that $\mathcal F_1 \subseteq \mathcal F_2 $. Can we always ...
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0answers
48 views

Expected frequency of most frequent die roll

Suppose we have an fair $m$-sided die, and we roll it $n$ times. What is the expected frequency $E(n, m)$ of the most frequently rolled face? If we fix $n$ we can calculate $E(n,m)$ like so. Let ...
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4answers
376 views

What is the expected number of coin tosses needed to obtain a head?

Due to my recent misunderstandings regarding the 'expected value' concept I decided to post this question. Although I have easily found the answer on the internet I haven't managed to fully understand ...
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3answers
76 views

Is it true in general that $E(1/X) = 1/E(X)$?

This concerns a discrete random variable $X$. I assume the relation doesn't hold in general, but I would like to prove this. I have tried to use the property that $$ E(g(X)) = \sum_x g(x)f(x) $$ ...
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1answer
54 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
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1answer
17 views

Why $E[X|\mathcal{G}]=X$ if $X$ is $\mathcal{G}$-measurable?

If $X$ is a $\mathcal{G}$-measurable random variable, why $E[X|\mathcal{G}] = X$? I know the intuition (basicly we're conditioning on the same informations on which $X$ is defined, $\sigma(X)$, we ...
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3answers
43 views

Calculating expected value for a Binomial random variable

How do you calculate $E(X^2)$ given the the number of trials and the probability of success? $E(X) = np$, then $E(X^2) = $? Do we have to draw up a table for $n=0,1,2,\ldots,n$ and then use the ...
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1answer
43 views

Why is this a geometric distribution?

For a random variable $X$, $$P(X = x) = (p-1)/p^{(x + 1)}$$ where $p$ is in $(1,\infty)$. Why is $X$ geometrically distributed? (and why would this make it true that $E[X] = 1 / (p - 1)$ ?) I know a ...
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1answer
38 views

How to apply the strong Markov property in this case?

I'm trying to understand the following proof: Theorem: Let $(X_n)$ be an irreducible $(\alpha, \mathbf p)$-Markov chain with a finite state space $S$. Then $(X_n)$ is positive recurrent. ...
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1answer
27 views

Equivalence of two formulas for variance and covariance

I know two formulas for variance: $$\operatorname{variance}(f) = \operatorname{expectation}((f(x) - \operatorname{expectation}(f^2(x)) \\ = \operatorname{expectation}(f(x)^2) - ...
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1answer
19 views

Centered independent increments process is a martingale

Let $(X_n)$ be an centered integrable process with independent increments (which as far as I understand means that $(X_{n+1}-X_n)_{n\in \mathbb N}$ is independent). While showing that $(X_n)$ is a ...
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2answers
26 views

Random Variable Worded Problem

I can figure out the basics to the question, that is the mean and variance of Y: E(Y) = 1-2p Var(Y) = 4p(1-p) I don't understand parts (i) and (ii). I dont understand the question itself, that ...
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0answers
7 views

Using EM versus estimating all parameters directly

My question is really a request for a clarification, and pertains to whether or not there are questions for which it is necessary to use EM or if it's more of a convenience. Let's presume we have n ...
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1answer
30 views

Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
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2answers
74 views

Brainteaser: Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. Expected number of games before one is bankrupt?

Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. What is the expected number of games they play before one is bankrupt? I have struggled at this for hours now with little ...
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97 views

About putting $n$ distinct balls into $n$ distinct boxes.

Let the balls be labelled $1,2,3,..n$ and the boxes be labelled $1,2,3,..,n$. Now I want to find, What is the expected value of the minimum value of the label among the boxes which are non-empty ...
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45 views

What is the expected value of the number of anchors of $S$?

For any subset $S\subseteq\{1,2,\ldots,15\}$, call a number $n$ an anchor for $S$ if $n$ and $n+ |S|$ are both elements of $S$. For example, $4$ is an anchor of the set $S=\{4,7,14\}$, since $4\in S$ ...
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1answer
39 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
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2answers
69 views

what is the difference between average and expected value?

I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the ...
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1answer
30 views

Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...
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1answer
37 views

how to compute $E[e^{a^2/2}N^2]$, $N$ is $\mathcal{N}(0,1)$

I have to show that $E[e^{(a^2/2)N^2}]=E[e^{(aNN')}]$ and tell for which values of $a$ these quantities are finite. $N$ and $N'$ are independent $\mathcal{N}(0,1)$ random variables I computed the ...
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1answer
15 views

expected value product dependent random variables

My question is strictly operative, if I have, for instance, two random variables $X$ and $Y$, $X$ is a $\mathcal{N}(m,\sigma^2)$ and $Y=e^{h(X-m)-1/2(h^2\sigma^2)}$. $E[Ye^X]$ is $\int y e^x p(x) ...
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1answer
66 views

Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
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1answer
52 views

$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
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1answer
42 views

Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...
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1answer
23 views

Convergence of expecations implies convergence of positive and negative parts?

If we have $E|X_n| \rightarrow E|X|$ does that imply \begin{equation} \lim_{n\rightarrow\infty} E X_n^\pm = X^\pm \end{equation} How about if we only have $EX_n \rightarrow EX$? Is this true in ...
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1answer
27 views

For which function $f$ is $1 \ll \sum_{i=1}^{n} i \cdot i^{-f(n)} \ll n$?

I am interested in the expected value of a power-law Distribution. I would like to let the Parameter $f(n)$ depend on $n$ for $n \rightarrow \infty$. And now I would like to determine $f(n)$ such ...
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1answer
24 views

Applying Markov's inequality to a sequence of random variables

Does the Markov inequality also work for infinite $a$ or only for constant $a$? More precisely: If $X(n)$ is a sequence of random variables and $f(n)$ is some sequence of numbers,is it allowed to ...
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2answers
87 views

Expected number of targets hit by 10 shooters, 2 bullets each, 20 targets

Consider a competition with 10 clay shooters. Each shooter has equal ability and they all use identical shotguns. Each shotgun is loaded with two bullets, therefore each shooter can shoot twice. ...
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1answer
15 views

What is the variance of multiple indicator random variables?!

Consider the following independent random variables $(V_1,V_2,V_3,\ldots,V_n)$ and a random variable $X$ as a function of these other random variables defined as follow on a set $A=(-\infty,x]$: $$ \ ...
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1answer
94 views

Expectation, random variable in exponent

Trying to solve a problem and got stuck trying to express this $E[2^{N(t)-N(s)}], (t>s)$ Where $N(t)$ is a Poisson process with unit rate, i.e. I'm trying to find $E[2^X]$ where $X$ has expected ...
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2answers
44 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
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1answer
18 views

When a r.v. admits mean and variance?

If $X$ is a r.v. on $(\Omega,\mathcal E, P)$ I have $E(X)=\int_\Omega X \, dP$ $\mathrm{Var}(X)=\int_\Omega (X-E(X))^2 \, dP=\int_\Omega X^2 \, dP - 2E(X) \int_\Omega X \, dP + E(X)^2 $ so $X ...
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1answer
32 views

Covariance of a function of random variables

I want to find the covariance $K_X(t,t')$ of the following signal $X(t)$: $X(t)=\sum\limits_{n=-\infty}^{+\infty} A_np(t-nT)$ where $ p(t) = \begin{cases} \ 1 & \text{if } 0<t\leq T/2 ...
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1answer
18 views

mgf and distribution where $E(X^m) = (m+1)!2^m$

From Hogg, McKean & Craig 6e; Problem 3.3.4 (I'm studying for a comprehensive exam in Math Stats, so this is almost like homework; I'd prefer answers that point me towards concepts I should ...
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1answer
38 views

Expected value and covariance of compound Poisson process

$Y_1,Y_2,...$ are independent random variables with a distribution identical to that of $Y$. $N(t)$ is a poisson process with parameter $\lambda$. $$X(t)=\sum\limits_{n=1}^{N(t)}Y_n$$ Find the ...
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0answers
45 views

An expectation inequality [closed]

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ...
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2answers
41 views

Question about computing expected value of the limit of a geometric mean of random variables

If I have the random variables $ X_{i} $ for $ i=1 \ldots N$ with the random variables being randomly selected integers from $1$ to $9$, how would I calculate the expected value of $$\lim_{N \to ...
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0answers
24 views

Geometric interpretation of expected value of a random variable

What's geometric interpretation of the expected value of a random variable? What I have understood so far can be elaborated as this: An expected value can be interpreted as the 'center of mass' of the ...
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1answer
35 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...
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39 views

numerical calculation of an integral

I am having trouble finding the solution of this numerically and wondered if I could get some tips so that I can: $$ \int\limits^1_{0}\left[\min(ax, b) - \min(a x, c)\right] dF(x; p, \rho)$$ (1) ...
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0answers
21 views

How to get a nice approximation of $f(N,s)=\sum_{k=0}^{N}{N \choose k}{k \choose s-k+N}$ when $N>>1$ and $|s|<<N$?

I need to approximate the above sum in order to calculate $\mathbb{E}(s^2)$, which is the expectation value determined by the probability density function $f$ and the position $s$. Any idea?
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2answers
46 views

Variance of number of tails in a coin-toss experiment

Let X be the random variable that equals the number of tails minus the number of heads when n fair coins are flipped. What is the variance of X? I've run a simulation and the answer seems to be n, ...
0
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1answer
38 views

$E(Y_i|X_i = 1)$ where $Y_i = X_i + U_i$ with $X_i$ being Bernoulli and $U_i$ being Normal

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
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1answer
34 views

Comparison of a functions with respect to two expectations

Suppose $X_i$ are i.i.d. nonnegative random variables and $w_i$ and $v_i$ are nonnegative weights. It is given that $\mathbb E \sum w_i X_i < \mathbb E \sum v_iX_i$. Then can we claim that ...
0
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3answers
167 views

Expected probablility in dice tossing

The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. When the dice is tossed, each face appears with ...