All Questions
Tagged with probability-theory expected-value
2,292
questions
2
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56
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Question about notation in Durrett's book
I have been studying Markov Chains through Rick Durrett's book, more precisely I was focusing on the Markov Property (Theorem 5.2.3 on page 276 of the book available at: https://services.math.duke.edu/...
- 31
3
votes
2
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If $f(x)$ is a pdf that is symmetric about $a$ (AKA $f(a+x) = f(a-x)$ for all $x \in \mathbb{R})$ then $EX = a$.
I'm looking for help for the problem in the title. If $f(x)$ is a probability density function (so it's nonnegative and its integral from $-\infty$ to $\infty$ is $1$) and is symmetric about $a$ (EDIT:...
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An application of Chebyshev association inequality?
Let $X$ be a r.v and let $f \geq 0 $ be a nonincreasing function, $g$ be a nondecreasing
real-valued function. Suppose $h\geq 0$ is a function such that $h(X)$ has finite expectation with $E[h(X)f(X)]...
0
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0
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Existence of $E(u)$; $E(u)=E(u^+-u^-)$
From chapter $IV.2$ of $$\textit{William Feller 'An Introduction
to Probability Theory
and Its Applications'; Vol.2}$$
$E(u)=\int_{\mathcal R^r}u(x)F\{dx\}$
I'm confused about the statement '$E(u)$...
- 883
2
votes
2
answers
80
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Generalization for Jensen's inequality for expected values
I'm trying to prove this. Let $\Phi$ be a convex function and X,Y random variables on the same probability space, where $Y \geq 0$ a.s. and $E(Y) = 1$. It holds $\Phi(E(XY)) \leq E(\Phi(X)Y)$.
What I ...
3
votes
1
answer
75
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Reverse engineering a property for Geometric Brownian Motion
During a lecture on financial mathematics we were given a side (non-homework) question to hone the SDE solving skills.
Setting
$\left(\Omega, \mathcal{F}, \mathbb{P},\left(\mathcal{F}_t\right)_{t \in[...
- 157
2
votes
1
answer
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Taking K elements from an infinite set sum, the expectation of getting a duplicate element exactly the Kth time
First, the probability of selecting $ k $ elements from an $ n $-element set, where the $ k $th selection is the first time a duplicate occurs, is given by:
$$
\frac{\binom{n}{k-1} (k-1)!(k-1)}{n^k}
$$...
- 21
1
vote
3
answers
139
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Representing a conditional expectation
Suppose I have a general random X, that is not necessarily continuous. It makes sense to me that I should be able to write:
$$E(X|X>c)=\dfrac{E(X\cdot 1_{X>c})}{P(X>c)}$$
If I assume that the ...
- 105
4
votes
1
answer
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How to find examples of $L^p$ converging random variables where a specified non-Lipschitz continuous function does not converge in $L^p$?
Background: I am preparing for a probability theory exam, and am struggling with a particular type of problem. The questions involve showing that if $X_n \xrightarrow{L^p} X$, then it does not ...
- 4,345
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38
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Follow-up to Expected-Value Dice question
Problem: Given that the first 6 occurs before the first 5, what is the expected number of rolls of a fair six-sided dice it will take to roll a 6 for the first time and stop?
This question was asked ...
- 167
1
vote
1
answer
61
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Chernoff bound of strings
Let $x\in(0,1)^*$ be a any binary string. Consider $m$ sample indices $i_1,i_2,\dots,i_m$ uniformly at random and independently of one another. Each $i_j$ is sampled uniformly at random from $[|x|]$ ...
user1290851
1
vote
1
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How can I show that $\mathbb{E}(T)=\frac{a^2}{\sigma^2}$?
Let $(X_i)_{i=1,2,...}$ be a sequence of iid random variables such that $\mathbb{P}(X_i=\pm 1)=\frac{1}{2}$ and with $\operatorname{Var}(X_i)=\sigma^2>0$. Set $S_t=\sum_{k=1}^t X_i$ for $t=1,2,...$ ...
- 2,935
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Layer cake representation and function with compact support
My question is related to the posts here and here, but my setup is slightly different.
For a real-valued random variable $X$, and a function $\varphi: \mathbb{R} \to \mathbb{R}$ that has support in ...
- 361
4
votes
5
answers
694
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Expected Number of Flips and Probability in a Coin Toss Experiment
I'm currently studying probability and I've come across a problem that I'm finding quite challenging. I would appreciate any help or guidance.
Problem Statement:
A fair coin is continually flipped ...
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1
vote
1
answer
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Expectation of $\mathbb{E}[\sum \delta_i / \sum T_i ]$
Let $T = \min\{X,C\}$, where $X \sim \mathsf{Exp}(\lambda)$ and $C \sim \mathsf{Exp}(\theta)$, $X$ and $C$ independent. Let $\delta$ be a random variable such that $\delta = 1$ if $X \leq C$ and $\...
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