All Questions
Tagged with expected-value independence
47
questions
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43
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How to calculate the expectancy of the ratio of non-independent random variables?
How can I calculate this expectancy:
$$
E \left [ \frac{\sum_{t=1}^T{Z_tX_t}}{\sum_{t=1}^T{Z_t^2}} \right ]
$$
where $Z_t \sim N(0,1)$ and $X_t \sim N(0,1)$ are independent? Any tricks? Is it ...
- 1,297
11
votes
3
answers
2k
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Are linear combinations of independent random variables again independent?
Let $X_1,X_2,\ldots,X_n$ be (iid) Random variables and define $Y_n:=\sum_{j=1}^na_jX_j$ with $a_j\in \mathbb{R}$, can we then say that the $a_jX_j$ are independent aswell. Can we express the MGF than ...
- 317
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32
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How to turn an expectation $E[A]$ into a conditional expectation, e.g., $E[A|B=1]$?
How can you turn an expectation $E[A]$ into a conditional expectation, e.g., $E[A|B=1]$, where:
A - continuous random variable, $A \in (-100, 1000)$
B - discrete r.v., $B \in {0, 1}$
A and B are ...
- 13
1
vote
1
answer
85
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Given independent random variables $X,Y$, and $M=\min(X,Y)$, what is $E(XM\mid Y=M)$?
Given independent random variables $X,Y$, and $M=\min(X,Y)$, what is $E(XM\mid Y=M)$ ?
The specific case I'm working on is assuming $X$ and $Y$ are exponential random variables with mean $\theta_X$ ...
- 11
1
vote
1
answer
161
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Calculate expectation of a function with two dependent random variables
Hi Cross Validated community,
My question has to do regarding expectation of a multiplication of two random variables that are dependent.
Assume there are two random variables, one discrete: $G \in \{...
- 13
-1
votes
1
answer
53
views
$cov(X,f(X))\neq 0$ and $E(X f(X))\neq 0$
Take a random variable $X$. Is it true that
(1) $cov(X,f(X))\neq 0$ for any function $f$?
(2) $E(X f(X))\neq 0$ for any function $f$?
I believe the answer to both questions is no. However, can you ...
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4
votes
1
answer
539
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some thought about independence and orthogonal, please comment on this if it's wrong
It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence.
In Statistics, the relation of ...
- 331
2
votes
1
answer
338
views
Expectation of the product of two independent random vectors and a positive-definite matrix
I am trying to compute the following: $\mathbb{E}[X^T\Omega^{-1}\epsilon]$, where $X$ is a random matrix, $\epsilon$ is a random vector, $\Omega$ is a real positive-definite matrix, and $\mathbb{E}[X^...
- 141
1
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47
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Expectation of products of (in)dependent random variables
Let $X_1, X_2, Y_1, Y_2$ be random variables and we are interested in $\mathbb{E}[X_1 X_2 Y_1 Y_2]$. How can we dissect this expectation if:
$X_1$ is independent of $X_2$ and $Y_1$ is independent of $...
- 941
0
votes
0
answers
117
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Find expected number of events happening given that events are dependent
So, my question frames like this:
There are n houses in row and there is a street light in between the houses. So, n-1 street lights.
The glowing of street light depends on both the immediate houses ...
- 103
5
votes
1
answer
366
views
Expectation of double quadratic form
I want to compute the following expectation $E(\hat{Y_k}'A\hat{Y_l}\hat{Y_k}'A\hat{Y_l})$ where $A$ is a symmetric non-random matrix and $E(\hat{Y_k}) = Y_k$, $E(\hat{Y_l}) = Y_l$. Additionally, $\hat{...
- 51
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149
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Expectation with respect to a product distribution
Let $\theta \in \Theta$ be a $d-$dimensional random variable.
Let $q$ be a distribution on $\Theta$ of the form $$q(\theta) = \prod_{i=1}^d q_i (\theta_i).$$
In other words $q$ is a product of ...
- 399
1
vote
0
answers
311
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Expected value of a product of two dependent random variables
Let me preface this by saying that I'm an engineer, and by no means a mathematician, so please excuse any mathematical "wrong-doing" in my explanation.
I have two vectors $V_1$ and $V_2$, ...
- 11
2
votes
1
answer
1k
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Expected Value of the Ratio of Independent Variables, E(X/(X+Y)) [duplicate]
If $X$ and $Y$ are independent random variables, is the following true? Is there an easy way to show this?
$$E\left[\frac{X}{X+Y}\right]=\frac{E[X]}{E[{X+Y}]}=\frac{E[X]}{E[X]+E[Y]}$$
If this is not ...
- 21
6
votes
1
answer
338
views
Independence of variables in expectation
I know that $X$ and $Y$ are independent, and have an expression $$E[I(Y>X)*I(X>2)].$$
Is the independence between $X$ and $Y$ enough to say that $$E[I(Y>X)*I(X>2)] = E[I(Y>X)]*E[I(X>...
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