All Questions
Tagged with probability-theory conditional-expectation
1,928
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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
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20
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Critical case of Galton-Watson Process
I'm reading the proof in Artheya, Branching Process (but I think this is a classical result) of the exponential limit law in the critical case of the Galton-Watson process i.e :
$\mathcal{L}(Z_n \vert ...
0
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0
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121
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On a conditional expectation property; substitution rule
For days now, I've been trying to prove the identity $$E(f(X,Y)\mid Y=y)=E(f(X,y)\mid Y=y).$$ I have found a couple of posts about this identity, mainly this one, and the more I think about this and ...
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Proof of a Conditional Expectation Result Using Truncation and Conditional Jensen's Inequality
Let $X, Y$ be integrable random variables. If with probability one
$E[X|Y] = Y$ and $E[Y|X] = X$, then $X = Y$ almost surely.
(Hint: first assume $X, Y$ are $\mathbb{L}^2$, and then use truncation and ...
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64
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Exercise on conditional expectation
i've got an exercise that i cannot solve properly. I am given $\Omega=\mathbb{R}$, $P[A]=\int_{A}f(x)dx$ for some density function $f: \mathbb{R}\to [0,\infty)$. I am asked to give an explicit formula ...
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2
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61
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Conditional Expectation of dependent normal distribution
Suppose we have $Z_0, Z_1, Z_2$ all standard normal distributed and independent. And $ X = c+aZ_0 + aZ_1$ and $Y=c+aZ_0+aZ_2$ for $a,c \ge 0$. Is there a way to calculate $E[Y|X]$ only using ...
- 615
2
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97
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Expected number of bonus wins for a specific slot machine game
I am analyzing a slot machine game for an assignment I was given.
There are two reels, one on the left and one on the right.
The probability of winning a spin is $p_L = 1/9$ for the left reel and $...
- 1,312
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Expectation of the process adapted to the filtration of the Wiener process
Suppose $\sigma_t$ is a stochastic process adapted to the filtration $\mathcal{F}_t$ generated by the Wiener process $W_t$.
I would like to know how to compute the following expectation:
$$E = \mathbb{...
- 367
1
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3
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139
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Representing a conditional expectation
Suppose I have a general random X, that is not necessarily continuous. It makes sense to me that I should be able to write:
$$E(X|X>c)=\dfrac{E(X\cdot 1_{X>c})}{P(X>c)}$$
If I assume that the ...
- 105
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Decomposing a general stopping time into stopping components
Let $(X_n)_{n \geq 0}$ be a discrete-time Markov chain taking values in a finite state space $S$, with transition matrix $P$. Let $(\mathcal F_n)_{n\geq 0}$ be the natural filtration and let $\tau \...
- 453
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Absolutely continuous random variables and conditioning
Suppose I have three random variables defined on a common probability space $(\Omega, \mathcal F, P)$. $X, Y$ taking values in $(\mathbb X, \mathcal X)$ and $Z$ taking values in $\mathbb Z, \mathcal Z)...
- 372
4
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2
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128
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Calculating a Conditional expectation
My question is the following. Given that we have $n$ i.i.d. random variables $X_1,...,X_n$ with distribution $f(x)=\frac{2}{\lambda^2}x\mathbf{1}_{[0,\lambda]}(x)$, where $\lambda> 0$ is some ...
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$\mathbb E(\max(X_1,...,X_{t+1})|\mathcal{F}_t)$ where the $X_i$ are iid uniform
Let $X_1,...,X_T$ be independent and identically distributed uniform random variables on $[0,1]$. Let $$M_t:=\max\{X_1,...,X_t\},$$
$L_t=M_t-ct$ for a $c>0$ and $L_0:=-\infty$. If $\mathcal{F}_t=\...
- 2,482
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Computing conditional expectation from a given random variable
Let $\Omega=\{a,b,c\}$, $\mathcal{F}=2^{\Omega}$ and $\mathbb{P}(a)=\mathbb{P}(b)=\mathbb{P}(c)=1/3$, so $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space.
Let $X$ be a random variable as ...
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Morris and Hirsch - $E(Y |X)$ linear function of $X$
Suppose that $X$ and $Y$ have a joint distribution with
means $μ_X$ and $μ_Y$ , standard deviations $σ_X$ and $σ_Y$ , and
correlation $ρ$. Show that if $E(Y |X)$ is a linear function of
$X$, then
$$E(...
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