Linked Questions
89 questions linked to/from What is the intuition behind conditional Gaussian distributions?
3
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What is the probability of $P(X>Y>0)$ where $X$ and $Y$ standard normal distribution with correlation $\rho$?
What I have done is use Cholesky decompostion to get $X = \rho X^{1} + \sqrt{1 - \rho^2}X^{2}$ and $Y = X^{1}$ where $X^{1}$ and $X^{2}$ are independent standard normal. Then
$$
\begin{aligned}
P(...
- 51
3
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1
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Multivariate normal distribution conditional variance independent of value of the given vector
$$\Sigma_{1,2}=\Sigma_{1,1}-\Sigma_{1,2}\Sigma_{22}^{-1}\Sigma_{2,1}$$
$$\mu_{1,2}=\mu_1+\Sigma_{1,2}\Sigma_{2,2}^{-1}(x_2-\mu_2)$$
This indicates that conditional variance of multivariate normal ...
- 565
1
vote
0
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764
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How to derive the covariance matrix from a rotated ellipse?
I'd like to derive the covariance matrix that defines a given ellipse.
Information I have:
length of major axis $\lambda_1$
length of minor axis $\lambda_2$
angle of rotation of the ellipse is $\...
- 23
0
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0
answers
577
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PCA with principal(): no rotation, varimax rotation, and oblimin rotation all give same results
I ran PCA with the principal() function of the psych package on some data with 2 variables and 15 observations . I ran PCA thrice, first with no rotation, then with the varimax rotation, and lastly ...
- 31
0
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0
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556
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What can regression slope tell us about the p-value and the overall utility of the model?
I am in the process of building a strong understanding of what I believe is the conceptual basis of linear regression.
One thing that I am struggling with is what is the relationship between ...
- 11
2
votes
3
answers
324
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Conditional expectation given sum of weighted average
Suppose X, Y are i.i.d standard normal (mean 0, standard deviation 1) random variables, a, b, c are constant scalars.
$$Z = a X + b Y$$
How to express $E[X|Z=c]$ using $a,b,c$?
- 130
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512
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Draw line through 2d density plot
have a large dataset of gene expression from ~10,000 patient samples (TCGA), and I'm plotting a predicted expression value (x) and the actual observed value (y) of a certain gene signature. For my ...
- 11
2
votes
2
answers
234
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How can PCA maximise variance after I standardise all predictor variance = 1?
I have been reading about Principal Components Analysis, and I think it is in general trying to extract as much "variance" out of the predictors $ \vec{X} = (X_1, X_2, ..., X_n)$ by ...
- 619
2
votes
3
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Rationale for elliptical region of correlations in gene expression data
I am analysing an RNA seq data set and I am trying to look at correlation between expression values of significant genes in 4 different biological duplicates and their clinical parameters.
Here, I ...
2
votes
1
answer
276
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Calculate number of "exceptions" to correlation
My question: Is there a formula to calculate analytically the expected number of "exceptions" which the simulator generates, as explained in the passage below?
For most of my adult life I've been ...
0
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0
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365
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Computing principal components of a bivariate normal distribution
Given a bivariate normal distribution $N(0, \Sigma)$ with $\Sigma=\begin{pmatrix}1 & \rho \\ \rho & 1 \end{pmatrix}$ and $\rho > 0$ I want to compute the principal components vector $Y$ ...
- 101
1
vote
1
answer
247
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conditional expectation of iid $X,Y$ cubic sum
Let $X$ and $Y$ i.i.d standardized normally distributed random variables.
Calculate the conditional expectation of :
$$ \mathbb{E}[(X+Y)^{3} | \mathscr{G}] $$
where $\mathscr{G} = \sigma(X)$ ($\sigma$...
user310375
1
vote
0
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314
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extract distribution of y for point x given lm of y~x
If I have two variables x and y that have a linear relationship e.g. using data from the mtcars package and R code
...
- 105
5
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0
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264
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Did Auguste Bravais really derive the mathematical definition of Pearson's product-moment correlation coefficient?
The wikipedia pages on Auguste Bravais,Karl Pearson, the Pearson correlation coefficient,and Francis Galton all cite the following book:
Bravais, A (1846). "Analyse mathématique sur les ...
- 9,680
2
votes
1
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309
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Conditional distribution of multivariate cauchy distribution
In the example of multivariate normal distribution,
$$
\begin{bmatrix}
\mathbf{x}_1 \\
\mathbf{x}_2
\end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix}
\mu_1 \\
\mu_2
\end{bmatrix}, \begin{...
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