All Questions
Tagged with expected-value random-variable
243
questions
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0
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34
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I don't think this is conditional dependence, so what is it?
I am looking for the name of the following phenomenon. There are three random variables, $X,Y,Z$. We have $P(X,Y) \neq P(X)P(Y)$ and $P(Y,Z) \neq P(Y)P(Z)$. In other words, $X$ and $Y$ are dependent, ...
- 111
5
votes
1
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104
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$E[(X+Y)^{a}] > E[(X)^{a}]$?
Assume I have two strictly positive i.i.d. random variables, $X$ and $Y$. Under what conditions is the following inequality true?
$$E[(X+Y)^{a}] > E[(X)^{a}], \hspace{2mm} a \in (0,1)$$
Should have ...
- 263
0
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0
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47
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Expectation of $u^\top(u+Ax)$, when $A$ and $u$ are nonlinear functions of $x$
Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that
$$u = s-y \\
v =(\operatorname{diag}(s) - ss^\top)x$$
I am interested in the inequality $u^\top (u ...
- 101
2
votes
1
answer
132
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Expected value of the product of three random variables
For two dependent random variables we have:
$$Cov[X, Y] = E[XY] - E[X]E[Y]$$
So that $E[XY] = E[X]E[Y] + Cov[X, Y]$
In case of three arbitrarily correlated random variables $(X, Y, Z)$, is it possible ...
- 725
0
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0
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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
1
vote
1
answer
80
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properties of a expectation for a non-negative random variable
Say I have a non-negative discrete random variable $X$ (values of $X$ can be mapped to integers $(0, 2^n -1)$ for $n \in \mathbb{Z}$) and an associated distribution $P(X)$. Given a non-negative scalar ...
- 13
1
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2
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107
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The training error of best hypothesis
Let $\mathcal{X}$ and $\mathcal{Y}$ denote the domain set and label set respectively. Also let $\mathcal{D}$ be a distribution over $\mathcal{X}$ and $f:\mathcal{X} \to \mathcal{Y}$ be the true ...
- 67
1
vote
0
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57
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Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
- 345
0
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47
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Order of centroids across two independent random variables
Say I have a function $q_{D, \Theta}: \mathcal{X} \to \mathcal{Y}$ that depends on independent random variables $D$ and $\Theta$.
I want to consider "centroids" of $q$ with respect to ...
- 339
3
votes
1
answer
119
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Expectation of product of sample averages
I have a bunch of iid random variables $X_i\sim q$ and I have defined other random variables $A_i = a(X_i)$ and $B_i = b(X_i)$. Then I bumped into the following expression
$$
\begin{align}
\mathbb{E}\...
- 2,236
1
vote
1
answer
105
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Expectation of the reciprocal of a standard normal random variable [duplicate]
If $\mathbf{X} \sim_{iid} \mathcal{N}(\mu, 1)$ then we know that the sample mean $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, how would we show that $$\mathbf{E}\left(\frac{1}{\bar{X}}\right) = \infty $$ and ...
- 41
1
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1
answer
164
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Show that for random variable $X$ with $N = \{1, 2, \ldots \}$, $E(X) = \sum_{n = 1}^\infty P(X \geq n)$ [duplicate]
Prove that for random variable with natural numbers from 1 to infinity the expected value $E(X)$ is equal to $\sum_{n = 1}^\infty P(X \geq n)$. Is this the mathematically correct way to prove it? And ...
- 45
0
votes
1
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113
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"Almost surely" used in an expectation
Let $(\mathsf{X}, \mathcal{X})$ be a measurable space, $\pi(dx)$ be a probability measure on it, and $K:X\times\mathcal{X}\to[0, 1]$ be a Markov kernel. I have the following property
$$
\int K(x, A) \...
- 657
1
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2
answers
228
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Understand the linearity of the expected value operator
Let Z = X + Y. The linearity of $\mathsf E$ implies that $\mathsf E [Z] = \mathsf E [X] + \mathsf E [Y]$. The left-hand side should be $\int (x+y)p_{x+y}(x+y)d(x+y)$. The right-hand side should be $\...
- 11
3
votes
1
answer
134
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Why do we weight the surprisals by the probabilities when computing the entropy?
Let's say that a random variable $X$ takes the values $(x_1, x_2, \dots, x_n)$ over the sample space $\Omega$.
As far as my understanding goes, the entropy of a given variable is meant to give an ...
- 245