15 votes
Accepted

Is calculating skewness necessary before using the z-score to find outliers?

The use of the phrase "the z-score method" in your title seems to assume that I should know of some method by that name. If we are to assume something specific from it, it would be best to ...
Glen_b's user avatar
  • 281k
15 votes
Accepted

Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)

The fact that the distribution of $X-Y$ has mean $0$ does NOT imply that $\mathbb{P}(X-Y<0)$ and $\mathbb{P}(X-Y>0)$ are equal. So for the last step it is not enough that $X-Y$ has mean $0$, you ...
user133281's user avatar
14 votes

Why does dbeta not sum to 1?

The relevant property of a probability density is not that it sums (for evaluation on some particular $x$ values) to one, but that it integrates to one. If you evaluate a density $f$ at $x$ values ...
Stephan Kolassa's user avatar
13 votes
Accepted

What distribution do I get when I square numbers from a normal distribution and add them together?

If $z$ has a standard normal distribution (mean of zero and variance of one), then $z^2$ has a chi-squared distribution with 1 degree of freedom. Furthermore, the sum of $x$ and $y$ from chi-squared ...
Gregg H's user avatar
  • 5,424
13 votes
Accepted

Does the PDF $\exp\left(-\frac{x^2}{2}\right) \cosh(\gamma x)$ have a name?

Since $$\cosh(\gamma x)=\dfrac{e^{\gamma x}+e^{-\gamma x}}{2}$$ one can rewrite the density as $$ p(x)\propto \exp(-{x^2}/{2}) \{e^{\gamma x}+e^{-\gamma x}\}\\ \propto e^{-x^2/2+\gamma x}+e^{-x^2/2-\...
Xi'an's user avatar
  • 104k
12 votes

Estimating a "most likely" distribution from min, max, mean, median, standard deviation

Obviously as a starting point to this kind of question, we must recognise that there are may possible probability distributions on the relevant support that have the same (finite) set of moments. ...
Ben's user avatar
  • 123k
12 votes

Can I assume normal distribution?

You can't assume a normal distribution of returns of financial assets to calculate the VaR. VaR is a measure of tail risk, and many financial assets have fat/heavy tails. It is very dangerous to make ...
Aksakal's user avatar
  • 60.9k
12 votes
Accepted

Can I use different transformations on features to ensure my data follow Gaussian Distribution

The Assumptions of Normality Regression does not require normality of either the predictors or the outcome. The typical assumption is that the residuals are normally distributed. Even this on its own ...
Shawn Hemelstrand's user avatar
11 votes

Conditional Expectation of Product of Normals given a Linear Combination

Comment on your attempt: the idea looks great but unfortunately $\xi - \eta \perp \xi + \eta$ of course does not imply $\xi - 2\eta \perp \xi + 2\eta$. However, the "product-to-sum" ...
Zhanxiong's user avatar
  • 17.4k
11 votes

Frobenius norm of a product of Gaussian matrices

Let $X$ be $d\times d$ a random matrix with iid $\mathcal N(0, 1/d)$ elements. Let us first show that $$\| XX^\top \|^2 \approx 2d.$$ Induction base We want to compute expectations of squared elements ...
amoeba's user avatar
  • 104k
10 votes
Accepted

How to show that $\{N(\theta,1):\theta \in \Omega\}$ is not a complete family of distributions when $\Omega$ is finite?

For notational convenience, let $\varphi_{\theta}(x)$ denote the density of a $N(\theta, 1)$ random variable. One way of viewing this problem is that the condition \begin{align*} \int_{-\infty}^\infty ...
Zhanxiong's user avatar
  • 17.4k
9 votes
Accepted

Expectation of a function of the sample covariance matrix

This is a quite challenging and interesting problem as it calls for many classical results of multivariate Gaussian distribution and technical matrix operations. To begin with, note that since $\hat{\...
Zhanxiong's user avatar
  • 17.4k
9 votes

Why does dbeta not sum to 1?

d* functions represent proportions only with a discrete response. In fact, your dnorm example just happens to sum to one, but <...
Alex J's user avatar
  • 2,011
9 votes

Why does dbeta not sum to 1?

dpois is the probability mass function (pmf) a discrete Poisson distribution that can take integer values in $(0, \infty)$. If you sum over the probabilities for ...
Björn's user avatar
  • 31.5k
9 votes
Accepted

Correlation between absolute deviation and standard deviation

We have $X \sim \mathrm{Unif}(a,b)$, with $a,b>0$, and $Y|X=x \sim \mathcal{N}(0, x^2)$. Notice that we can write $Y=Z X$, where $Z$ is standard normal (and $Z \perp \!\!\! \perp X$). The ...
Doctor Milt's user avatar
  • 2,682
9 votes

Best estimator of the mean of a normal distribution based only on box-plot statistics

An exact answer will be difficult, so first I will look at asymptotic theory. Answers from that could be tested by simulation, comparing it to a maximum likelihood estimator computed by maximizing an ...
kjetil b halvorsen's user avatar
8 votes
Accepted

Why do we need normality test if we already have CLT?

1. The CLT certainly doesn't solve all problems. For example: (a) There are distributions for which the CLT doesn't hold. Here's an example: This density is a mixture of a symmetric 4-parameter beta ...
Glen_b's user avatar
  • 281k
8 votes

Relationship between z-score and the normal distribution

There is no relationship. The (sample) z-score is defined as $$ z_i = \frac{ x_i - \bar x } {s} $$ where $i$ indexes observations $\{x\}$, $\bar x$ is the sample mean, and $s$ is the sample standard ...
Alex J's user avatar
  • 2,011
8 votes
Accepted

Probability that $k$ variables of multivariate Gaussian are positive

Suppose $(X_1,X_2,\ldots,X_n)$ has a standard multivariate normal distribution with common correlation $\frac12$. Then you have the following equality in distribution: $$(X_1,X_2,\ldots,X_n) \stackrel{...
StubbornAtom's user avatar
8 votes
Accepted

Finding $\mathbb E(Y_1^2Y_2^2)$ when $(Y_1,Y_2)$ is normal

Since all Gaussian variables have a kurtosis of $3,$ any Gaussian variable with variance $\sigma^2$ has a central fourth moment of $3\sigma^4.$ In particular, when $Z$ is a zero-mean Gaussian, $$E[Z^...
whuber's user avatar
  • 321k
8 votes

Finding $\mathbb E(Y_1^2Y_2^2)$ when $(Y_1,Y_2)$ is normal

Use the Characteristic function. It is well known that, $$\varphi_Y(s) := \mathbb E\left[e^{is^\intercal Y}\right] = e^{-\frac12s^\intercal \Sigma s}.$$ So \begin{align} \frac{\partial^2 \varphi_Y}{\...
Kroki's user avatar
  • 270
8 votes
Accepted

What is the distribution of the predictions in linear regression?

Assume a traditional linear regression and that the data follow the model, i.e, given $X$ the corresponding value $Y|X$ will be distributed as $\mathcal{N}(X \beta, \sigma^2)$ for some true values of $...
user9794's user avatar
  • 216
8 votes

Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)

Your reasoning seems right. You can always do some simulation to check if your answer is correct. ...
Peter Flom's user avatar
  • 117k
8 votes
Accepted

Normality test using normal Q-Q plot and histogram

While I agree with whuber's comment that this is unlikely to be useful, and while you don't say why you are doing this, I think that, nevertheless, something useful can be said. Of course, if you have ...
Peter Flom's user avatar
  • 117k
7 votes
Accepted

How to determine the correlation between two normal random variables conditioned on their sum being negative?

The conditional expectations of $X$ and $Y$ are obviously equal. Moreover, because $(X+Y)/\sqrt 2$ has a standard Normal distribution, its conditional expectation is the negative of $-|Z|$ where $Z$ ...
whuber's user avatar
  • 321k
7 votes

normal distribution in machine learning

It is a bit hard to understand what the $I$ is really, but in general this notation is used to designate a squared matrix with $1$ on the diagonal and $0$ elsewhere. $I$ is called the identity matrix. ...
lulufofo's user avatar
  • 447
7 votes
Accepted

How to estimate bias-corrected variance of a half-normal distribution?

If $Y_1, \ldots, Y_n \sim \mathcal{N}(0, \sigma^2)$ and $X_i=|Y_i|$ for $i=1,\ldots,n$, we say that $X_1,\ldots,X_n$ is a random sample from a half-normal distribution with scale parameter $\sigma$. ...
Doctor Milt's user avatar
  • 2,682
7 votes

Finding $\mathbb E(Y_1^2Y_2^2)$ when $(Y_1,Y_2)$ is normal

You can use the conditional distributions, which are univariate normal. For example, $Y_2$ given $Y_1$ is normal with mean $$E\left[Y_2\mid Y_1\right]=\frac{\sigma_{12}}{\sigma_{11}} Y_1$$ and ...
StubbornAtom's user avatar
7 votes
Accepted

What is the substantive meaning of one statistical test is more powerful than another?

If a test has any power and always returns $p=1.0$ that implies that the sample size is too small to be doing statistical testing. Power of competing tests is compared at the null, to show that the ...
Frank Harrell's user avatar
7 votes
Accepted

Bivariate normal covering circles and ellipses

What you want is the radius $r$ such that the cdf $P(\sqrt{X^2+Y^2}\leq r)=\alpha$ for some alpha. It is clear that $P(\sqrt{X^2+Y^2} \leq r)=P(X^2+Y^2 \leq r^2)$, and it is easier to work with the ...
dherrera's user avatar
  • 1,248

Only top scored, non community-wiki answers of a minimum length are eligible