All Questions
Tagged with expected-value normal-distribution
213
questions
2
votes
0
answers
94
views
$E(XY)$ for a truncated bivariate normal
If $(X, Y)$ follows a bivariate Gaussian distribution with mean ${\bf \mu}$ and covariance ${\bf \Sigma}$ with truncation bounds $(a_x, b_x, a_y, b_y)$, can we compute $E(XY)$ in closed form? If not, ...
- 8,392
0
votes
0
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37
views
Ratio of Normal Distributions [duplicate]
Suppose I have two independent random variables, $X \sim N(\mu_1,\sigma_1^2)$ and $Y \sim N(\mu_2,\sigma_2^2)$ with $\mu_1,\mu_2 > 0$.
How can I compute/estimate
$$ \mathbb{E}\left[\left\lvert \...
- 119
6
votes
1
answer
73
views
$E[W\otimes W]$ for Wishart R.V. $W$
What is the value of $E[W\otimes W]$ for Wishart R.V. $W$?
$\otimes$ refers to Kronecker product
I found related formula for $E[WAW]$ on page 467 of Seber's Matrix handbook, wondering if $E[W\otimes W]...
- 6,307
3
votes
1
answer
100
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Expectation of the Gaussian likelihood
I'm working on a challenging machine learning problem, where I need to find the expectation of the likelihood of one Gaussian, given the parameters of another. Apologies if any of the notation is ...
1
vote
1
answer
105
views
Expectation of the reciprocal of a standard normal random variable [duplicate]
If $\mathbf{X} \sim_{iid} \mathcal{N}(\mu, 1)$ then we know that the sample mean $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, how would we show that $$\mathbf{E}\left(\frac{1}{\bar{X}}\right) = \infty $$ and ...
- 41
2
votes
1
answer
116
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Moments of sum of squares of independent gaussians $X_i \sim \mathcal{N}(\mu_i,\sigma^2_i)$, or $||X||^2$
Say that we have $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. Is there some formula to calculate analytically the expected value of the sum $S = \sum_i^n X_i^2$?. This is equivalent to computing $\...
- 1,328
3
votes
1
answer
87
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Given $r\gt 0$, how to get $\mu_r = E[|U|^r]$ where $U\sim N(0,1)$?
Given a standard normal random variable $U$ , is there a general formula to compute the expected value of the absolute value of $U$ to any power?
For example given a non negative constant $r$ (i.e $r\...
- 115
0
votes
0
answers
54
views
Variance of powers of a standard normal random variable
To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market ...
- 139
0
votes
0
answers
30
views
Expectations with respect to affine transformation of a log-normal distribution
Let $X$ be a log-normal distribution and consider $Y=aX+b$ for some $a,b>0$.
I would like to know if one can compute
$$\mathbb{E}[\log(Y)]$$
This would be very easy if it was $b=0$, since in this ...
0
votes
1
answer
225
views
Expected value of Truncated Normal Distribution [duplicate]
For the truncated normal distribution below:
$$
{f_X(x; σ, a, b)} = \frac{1}{\sigma}\frac{φ(\frac{x-µ}{σ})}{Φ(\frac{b-µ}{σ})-Φ(\frac{a-µ}{σ})}
$$
$$
a = 1; b = ∞; σ = 2
$$
I need to calculate the ...
- 109
9
votes
5
answers
279
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Finding $\mathbb E(Y_1^2Y_2^2)$ when $(Y_1,Y_2)$ is normal
Let $$Y =\begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix}
\sim N(0, \Sigma) \quad \Sigma=\begin{pmatrix} \sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22} \end{pmatrix}$$
Show that $$\mathbb E(Y_1^...
- 445
4
votes
1
answer
102
views
Expectation of two Quadratic form
Assume
$\mathbf{h} \in C^{N \times 1}$ is a Gaussian vector with zero mean and Covariance matrix $\mathbf{R}$.
Also $\mathbf{A} \in C^{N \times N}$ is a deterministic diagonal matrix. In this case, ...
6
votes
1
answer
110
views
Expected Value of the Difference Between the Powers of Two Randomly Selected Numbers From a Standard Normal Distribution
My question is similar to (and an extension of) this one.
I select two values ($i$ and $j$) at random from a standard normal distribution. What is the expected value of $|x_i^n-x_j^n|$ for all integer ...
- 247
2
votes
1
answer
193
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Expectation of the absolute value of the product of correlated jointly gaussians?
I am reading the Performer paper https://arxiv.org/abs/2009.14794. To understand their ReLU kernel used to approximate softmax attention, I need to evaluate $\mathbb{E}[ReLU(x^T w) \cdot ReLU(y^T w)]$ ...
- 23
0
votes
0
answers
30
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Computing the expected value of a new Normal Random Variable (transformation)
I have the following exercise to do:
Let X be a normally distributed variable with mean μ and variance σ^2, i.e. X∼N(μ,σ^2). Define a new random variable to be Z=X^2−X. Compute the Expected Value of Z
...
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