# Tag Info

1 vote

• 56.5k
1 vote
Accepted

### How to test for normality on paired samples between two treatments when the number of observations per treatment is unequal?

The number of pokes in 1 hour is count data, therefore I would analyze it with a GLM for a distribution of the Poisson family. You have 2 measurements on the same mouse, therefore mouse identity ...
• 76

### How to test for normality on paired samples between two treatments when the number of observations per treatment is unequal?

There is no reason to expect normality. Think about using the rank difference test or its regression model generalization described here. You have to acccount for a poke being a response variable ...

### How can we maintain asymptotic normality with slight change?

I won't solve this for you, but the way I'd go on about it is that (just to use an example) I'd assume $a_n\not\to 1$ (or any other of the required conditions not fulfilled). This means that there is ...
• 19.4k
1 vote

### Is it reasonable to use the Kolmogorov-Smirnov test to assess the normality of a random variable?

No one here has mentioned the Liliefors test. The Liliefors test is like the Kolmogorov–Smirnov test, comparing the empirical c.d.f. with the c.d.f. of the normal distribution with the same mean and ...
• 8,561
1 vote

### Which is largest, of a bunch of normally distributed random variables?

We have \begin{align} \mathbb{P}(\mathcal{E}) &= \mathbb{P}(\bigcap_{i=1}^n \{X_0 \ge X_i \}) = \mathbb{P}(\bigcap_{i=1}^n \{X_0 -X_i \ge 0 \}) \end{align} Let us denote $Z_i = X_0 - X_i$ ...
• 188
1 vote

### How to calculate $\int_{-\infty}^{\infty} \Phi(y+c)^{k-1}d(\Phi(y))$?

This integral is equal to $\mathbb{E}(\Phi^{k-1}(Y+c))$ for $Y$ follows the distribution $\mathcal{N}(0,1)$. By applying this answer with $a = 1, n = k-1$, we can have a closed-form expression for ...
• 188

### Expected value of softmax transformation of Gaussian random vector

Late to the party, but there's another (perhaps less complicated) avenue to calculate the value of $s(x)$, and that's taking the average of $s(x)$ over a very large number of runs.
• 176
1 vote
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### Calculating the mean and sd of a lognormal distribution from the log mean and log 5th percentile

Let's call the (as yet unknown) mean and standard deviation of the underlying normal $\mu$ and $\sigma$, so $X \sim N(\mu,\sigma^2)$, and the (known) mean of the lognormal $m_Y$ with $5$th percentile ...
• 36.7k
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### Data transformation to normal distribution in analysis of variance (ANOVA)

There are various tests of normality. I'm not sure I'd call them "rigorous" and I'd advise against their use in this context. In any case, ANOVA does not require normal data, one of its ...
• 105k
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### Poisson Regression and Linear Regression give the same error

Your expectation is wrong. It's all about evaluation metrics and loss functions: Poisson regression minimizes the average unit Poisson deviance. lm() minimizes the ...
• 11.5k

### What is the probability of $P(X>Y>0)$ where $X$ and $Y$ standard normal distribution with correlation $\rho$?

Given that the joint density of $(X, Y)$ is $f(x, y) = \frac{1}{2\pi\sqrt{1 - \rho^2}}\exp\left(-\frac{x^2 + y^2 - 2\rho xy}{2(1 - \rho^2)}\right)$, the probability of interest is \begin{align} P(X &...
• 14.6k
Accepted

### All uncorrelated marginals are independent: Only for joint Gaussian?

The answer is yes. This is a consequence of the Darmois-Skitovich theorem. It states that if $X_1,\ldots,X_p$ are independent and $\alpha_i,\beta_i\neq 0$ for all $i\in\{1,\ldots,p\}$, then $\alpha^TX$...
• 213