# Tagged Questions

The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is often used as a reference against which other distributions are compared.

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### A Probability distribution value exceeding 1 is OK?

On the Wikipedia page about naive Bayes classifiers, there is this line: $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve ...
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### What is the most surprising characterization of the Gaussian (normal) distribution?

A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: $$\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ or its characteristic function. As recalled in this ...
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### Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?

Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance ...
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### T-test for non normal when N>50?

Long ago I learnt that normal distribution was necessary to use a two sample T-test. Today a colleague told me that she learnt that for N>50 normal distribution was not necessary. Is that true? If ...
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### Bottom to top explanation of the Mahalanobis distance?

I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still ...
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### Confidence Interval for variance given one observation

This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory": Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters ...
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### Percentage of overlapping regions of two normal distributions

I was wondering, given two normal distributions with std1,mean1 and std2,mean2, how can I calculate the percentage of overlapping regions of two distributions. I suppose this problem has a specific ...
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### What is the distribution of the euclidean distance between two normally distributed random variables

Assume you are given two objects whose exact locations are unknown, but are distributed according to normal distributions with known parameters (e.g. $a \sim N(m, s)$ and $b \sim N(v, t))$. We can ...
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### How can I estimate the probability of a random member from one population being “better” than a random member from a different population?

Suppose I have samplings from two distinct populations. If I measure how long it takes each member to do a task, I can easily estimate the mean and variance of each population. If I now hypothesise ...
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### What is the importance of the function $e^{-x^2}$ in statistics?

In my calculus class , we encountered the function $e^{-x^2}$, or the "bell curve", and I was told that it has frequent applications in statistics. Out of curiosity, I want to ask: Is the function ...
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### What tests do I use to confirm that residuals are normally distributed?

I have some data which looks from plotting a graph of residuals vs time almost normal but I want to be sure. How can I test for normality of error residuals?
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### Approximate order statistics for normal random variables

Are there well known formulas for the order statistics of certain random distributions? Particularly the first and last order statistics of a normal random variable, but a more general answer would ...
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### How to compute the confidence interval of the ratio of two normal means

I want to derive the limits for the $100(1-\alpha)\%$ confidence interval for the ratio of two means. Suppose, $X_1 \sim N(\theta_1, \sigma^2)$ and $X_2 \sim N(\theta_2, \sigma^2)$ being independent, ...
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### Error in normal approximation to a uniform sum distribution

One naive method for approximating a normal distribution is to add together perhaps $100$ IID random variables uniformly distributed on $[0,1]$, then recenter and rescale, relying on the Central Limit ...
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### How to formally test for a “break” in a normal (or other) distribution

It frequently comes up in social science that variables that should be distributed in some way, say normally, end up having a discontinuity in their distribution around certain points. For instance, ...
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### Hypothesis testing on the inverse covariance matrix

Suppose I observe i.i.d. $x_i \sim \mathcal{N}\left(\mu,\Sigma\right)$, and wish to test $H_0: A\$vech$\left(\Sigma^{-1}\right) = a$ for a conformable matrix $A$ and vector $a$. Is there known work ...
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### Why it is often assumed Gaussian distribution?

Quoting from a Wikipedia article on parameter estimation for a naive Bayes classifier: "a typical assumption is that the continuous values associated with each class are distributed according to a ...
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### Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s

I'm working with two independent normal distributions $X$ and $Y$, with means $\mu_x$ and $\mu_y$ and variances $\sigma^2_x$ and $\sigma^2_y$. I'm interested in the distribution of their ratio ...
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### Why would all the tests for normality reject the null hypothesis?

The Kolgomorov-Smirnov test, Shapiro test, etc.... all reject the hypothesis that a distribution is normal. Yet when I plot the normal quantiles and and histogram, the data is clearly normal. Maybe ...
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### Distribution of extremal values

If an item follows normal distribution, average also follows normal distribution. What about minimum and maximum?
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### Randomized trace technique

I've met the following randomized trace technique in M. Seeger, “Low rank updates for the Cholesky decomposition,” University of California at Berkeley, Tech. Rep, 2007. ...
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### Should I use a binomial cdf or a normal cdf when flipping coins?

A coin needs to be tested for fairness. 30 heads come up after 50 flips. Assuming the coin is fair, what is the probability that you would get at least 30 heads in 50 flips? The right way to do ...
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### Properties of bivariate standard normal and implied conditional probability in the Roy model

Sorry for the long title, but my problem is quite specific and hard to explain in one title. I am currently learning about the Roy Model (treatment effect analysis). There is one derivation step at ...
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### Generate random correlation matrix that has approximately normal entries

I would like to generate a random correlation matrix such that the distribution of the entries (except 1) look approximately like normal. The motivation is this. For a set of n time series data, the ...
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### When data has a gaussian distribution, how many samples will characterise it?

Gaussian data distributed in a single dimension requires two parameters to characterise it (mean, variance), and rumour has it that around 30 randomly-selected samples is usually sufficient to ...
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### Gaussian Like distribution with higher order moments

For the Gaussian distribution with unknown mean and variance, the sufficient statistics in the standard exponential family form is $T(x)=(x,x^2)$. I have a distribution that has ...
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### How can I find the standard deviation of the sample standard deviation from a normal distribution?

Forgive me if I've missed something rather obvious. I'm a physicist with what is essentially a (histogram) distribution centered about a mean value that approximates to a Normal distribution. The ...
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### Calculating percentile of normal distribution

See this Wikipedia page: http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Agresti-Coull_Interval To get the Agresti-Coull Interval, one needs to calculate a percentile of the ...
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### How is Poisson distribution different to normal distribution?

I have generated a vector which has a Poisson distribution, as follows: x = rpois(1000,10) If I make a histogram using ...
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### Sum of two normal products is Laplace?

It is apparently the case that if $X_i \sim N(0,1)$, then $X_1 X_2 + X_3 X_4 \sim \mathrm{Laplace(0,1)}$ I've seen papers on arbitrary quadratic forms, which always results in horrible non-central ...
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### Why does the McNemar's test use chi-square and not the normal distribution?

I just noticed how the non exact McNemar's test uses the chi square asymptotic distribution. But since the exact test (for the two case table) relies on the binomial distribution, how come it is not ...
I have random variables $X_0,X_1,\dots,X_n$. $X_0$ has a normal distribution with mean $\mu>0$ and variance $1$. The $X_1,\dots,X_n$ rvs are normally distributed with mean $0$ and variance $1$. ...
Based on a sample $(x_i) \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)$, how can you get an exact or a well-approximated upper tolerance bound (i.e. an upper confidence bound of a quantile of the ...