All Questions
135
questions
1
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$(X|Y=y)\sim N(y,y^2)$, $Y\sim U[3,9]$. Find $Var(X).$
$(X|Y=y)\sim N(y,y^2)$, $Y\sim U[3,9]$, where $N(y,y^2)$ is a normal distribution with mean $y$ and variance $y^2$, $U[3,9]$ is a uniform distribution on $[3,9].$
On this condition, find $Var(X).$
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- 903
0
votes
2
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499
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Conditional expectation conditioned both to a random variable and an event
Consider a uniform random variable $U$ in the interval $(0,1)$. Trivially we have that
$$\mathbb{E}[X|X<\tfrac{1}{2}]=\frac{1}{4}.$$
Now, based on my intuition, I would like to say that $(X|X<...
- 1,054
1
vote
1
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151
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A question about conditional expectation problem from Durrett
Let $X$ be a random variale such that $E(X \mid G)$ has the same distribution as $X$, $E|X|<\infty$. Prove that $$\text{sgn}(X) = \text{sgn}(E(X \mid G)).$$
The solution is given here where I ...
user621937
3
votes
2
answers
103
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Compute $\mathbb{E}\big[\exp(XY+X)\big]$ where $X, Y$ are independent uniformly distributed over $[0,1]$ r.v's.
Compute $\mathbb{E}\big[\exp(XY+X)\big]$ where $X, Y$ are independent uniformly distributed over $[0,1]$ r.v's.
I first computed $\mathbb{E}\big[\exp(XY+X)\mid X\big]$.
Because $X$ and $Y$ are ...
- 2,425
2
votes
1
answer
124
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Proving $\mathbb{E}(S_{\tau}|\tau)=\tau \mathbb{E}(\xi_1)$ and $\mathbb{E}S_{\tau}=\mathbb{E}\tau \mathbb{E}\xi_1$
Let $\xi_1, x_2,...,\xi_n$ and $\tau$ be independent random variables in $(\Omega, \mathcal{F}, \mathbb{P})$. $\xi_1, x_2,...,\xi_n$ are uniformly distributed and $\tau=\{0,1,2,...,n\}$. Let $S_{\tau}=...
- 337
0
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2
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120
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integrable random variables $X,Y$ such that $E(X|Y)=X$ a.s. and $E(Y|X)=Y$ a.s.
Let $X,Y$ are random variables such that $E(|X|)+E(|Y|)<\infty$, and the random variable $E(X|Y)=X$ a.s. and $E(Y|X)=Y$ a.s .
Then is it true that $X=Y$ a.s. ?
- 3,705
1
vote
1
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104
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Random variables $X,Y$ such that $E(X|Y)=E(Y|X)$ a.s.
Let $X$ and $Y$ be random variables such that $E(|X|), E(|Y|)<\infty$ and $E(X|Y)=E(Y|X)$ a.s. Then is it true that $X=Y$ a.s. ? If this is not true in general, what happens if we also assume $X,Y$ ...
- 3,705
1
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2
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57
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Conditional expectation of a sequence of random variables
$X_1 \sim \mathrm{Unif}(0,1)$
if $X_1=x_1$, $X_2 \sim \mathrm{Unif}(x_1,x_1+1)$
if $X_2=x_2$, $X_3 \sim \mathrm{Unif}(x_2,x_2+1)$
for $n \geq4$, $X_n$ is defined the same way. How do I calculate $E(...
- 235
0
votes
1
answer
59
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On the random variable which denotes the number of flips, in a coin toss, it takes to get a run of $n$ successive heads
Let the probability of obtaining head, when a coin is tossed, be $p$. Let the coin be tossed, many times, independently and
$X_n :=$the number of flips it takes to get a run of $n$ successive heads.
...
- 3,705
0
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1
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74
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Computing $E[X|XY]$ [closed]
Let $X$ and $Y$ be independent continuous random variables with densities $f_{X}(x)$ and $f_{Y}(y)$. Also, let $Z = XY$. How does one compute $E[X|Z]$?
- 2,885
2
votes
1
answer
114
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Conditional Probability Question for A Continuos Random Variable
Suppose $X$ is a continuous random variable with density $p$ with respect to the Lebesgue measure. According to Radon-Nikodym theorem $p$ is measurable itself so it can be seen as a random variable. ...
- 1,281
1
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1
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183
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Understanding of conditional expectation as a random variable
On page 3 of Knuth's Pre-Fascicle 5A (Mathematical Preliminaries Redux) of TAOCP, it reads:
... you might think of $E(X \mid Y)$ as a function of $Y$. Well, yes; but the best way to understand $E(X ...
- 3,707
2
votes
1
answer
57
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Finding conditional expectation $E[f|g]$ given $f(x):= x^2 /2$ and $g(x):=2(x-1/2)^2$
Let $([0,1],\mathscr B[0,1],Leb)$ be a probability space.
Define $f(x):= x^2 /2$ and $g(x):=2(x-1/2)^2$.
Find $$E[f|g], E[g|f]$$
What I have tried:
It is clear that $E[g|f]=g$ as $\sigma(f)=\mathscr ...
- 1,235
0
votes
1
answer
113
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Iterated expectation (V-statistic)
Suppose we have a function $g$ and two random variables $\tilde{X} = (\tilde{X}_1, \tilde{X}_2, \tilde{X}_3)$ and $X = (X_1, X_2, X_3)$ which are iid. Furthermore, $\tilde{X}_1, \tilde{X}_2, \tilde{X}...
- 59
1
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1
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176
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Basic Property of Conditional Expectation
Let $(\Omega, F, P)$ probability space. Let $B \subseteq F$. Let $X,Y \in \mathcal{L}^1(P)$
I know, that:
$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X|B]]$
But, is this legal?
$\mathbb{E}[X] = \mathbb{...
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