The normal, or Gaussian, distribution is a symmetrical bell-shaped curve that is defined by the mean and standard deviation. Parametric statistics tests require the population to be normally distributed. A sample distribution that is normally distributed is used to assume that the population ...
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3answers
3k views
A Probability distribution value exceeding 1 is OK?
On the Wikipedia page about naive bayes classifiers here there is this line "P(height|male) = 1.5789 (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)" ...
22
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12answers
2k views
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 ...
21
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2answers
1k views
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 Normal with their means and variance ...
18
votes
6answers
13k views
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 ...
16
votes
3answers
400 views
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 ...
15
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3answers
309 views
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 ...
15
votes
3answers
5k views
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 ...
14
votes
3answers
600 views
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 ...
14
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3answers
2k views
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 ...
13
votes
5answers
590 views
Comparing the variance of paired observations
I have $N$ paired observations ($X_i$, $Y_i$) drawn from a common unknown distribution, which has finite first and second moments, and is symmetric around the mean.
Let $\sigma_X$ the standard ...
11
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1answer
3k views
Deriving the conditional distributions of a multivariate normal distribution
We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into
$$\boldsymbol\mu
=
\begin{bmatrix}
...
11
votes
1answer
440 views
A hair dresser's conundrum
My hairdresser Stacey always puts on a happy face, but is often stressed about managing her time.
Today Stacey was overdue for my appointment and very apologetic. While getting my haircut I wondered:
...
11
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3answers
293 views
Estimating parameters for a spatial process
I'm given an $n\times n$ grid of positive integer values. These numbers represent an intensity that should correspond to the strength of belief of a person occupying that grid location (a higher value ...
10
votes
3answers
956 views
Evaluate definite interval of normal distribution
I know that an easy to handle formula for the CDF of a normal distribution is somewhat missing, due to the complicated error function in it.
However, I wonder if there is a a nice formula for ...
10
votes
3answers
2k views
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 ...
10
votes
3answers
992 views
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?
10
votes
5answers
4k views
How to sample from a normal distribution with known mean and variance using a conventional programming language?
I've never had a course in statistics, so I hope I'm asking in the right place here.
Suppose I have only two data describing a normal distribution: the mean $\mu$ and variance $\sigma^2$. I want to ...
10
votes
2answers
236 views
Finding number of gaussians in a finite mixture with Wilks' theorem?
Assume I have a set of independent, identically distributed univariate observations $x$ and two hypotheses about how $x$ was generated:
$H_0$: $x$ is drawn from a single Gaussian distribution with ...
9
votes
3answers
941 views
How to compute the probability associated with absurdly large Z-scores?
Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these ...
9
votes
1answer
175 views
Density of normal distribution as dimensions increase
The question I want to ask is this: how does the proportion of samples within 1 SD of the mean of a normal distribution vary as the number of variates increases?
(Almost) everyone knows that in a 1 ...
9
votes
1answer
506 views
Intuitive explanation of contribution to sum of two normally distributed random variables
If I have two normally distributed independent random variables $X$ and $Y$ with means $\mu_X$ and $\mu_Y$ and standard deviations $\sigma_X$ and $\sigma_Y$ and I discover that $X+Y=c$, then (assuming ...
9
votes
1answer
493 views
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 ...
9
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3answers
396 views
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, ...
9
votes
1answer
246 views
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 ...
9
votes
3answers
883 views
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 ...
8
votes
2answers
2k views
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, ...
8
votes
5answers
3k views
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 ...
8
votes
3answers
186 views
Distribution of extremal values
If an item follows normal distribution, average also follows normal distribution. What about minimum and maximum?
8
votes
1answer
299 views
Central limit theorem and the law of large numbers
I have a very beginner's question regarding the Central Limit Theorem (CLT):
I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for $n \to ...
8
votes
1answer
221 views
Determining true mean from noisy observations
I have a large set of data points of the form (mean, stdev). I wish to reduce this to a single (better) mean, and a (hopefully) smaller standard deviation.
Clearly I could simply compute $\frac{\sum ...
8
votes
3answers
225 views
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.
...
8
votes
1answer
260 views
Does the multivariate Central Limit Theorem (CLT) hold when variables exhibit perfect contemporaneous dependence?
The title sums up my question, but for clarity consider the following simple example. Let $X_i \overset{iid}{\backsim} \mathcal{N}(0, 1)$, $i = 1, ..., n$. Define:
\begin{equation}
S_n = \frac{1}{n} ...
8
votes
2answers
242 views
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 ...
8
votes
2answers
268 views
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 ...
8
votes
0answers
275 views
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 ...
7
votes
2answers
219 views
Which is largest, of a bunch of normally distributed random variables?
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 $1$. ...
7
votes
2answers
224 views
Tolerance bound for a normalized variable
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 ...
7
votes
4answers
654 views
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 ...
7
votes
1answer
188 views
“Since $x$ is near-gaussian, its PDF can be written as…”
Short question: Why is this true??
Long question:
Very simply, I am trying to figure out what justifies this first equation. The author of the book I am reading, (context here if you want it, but ...
7
votes
2answers
374 views
Estimating parameters of a normal distribution: median instead of mean?
The common approach for estimating the parameters of a normal distribution is to use the mean and the sample standard deviation / variance.
However, if there are some outliers, the median and the ...
7
votes
2answers
2k views
Mahalanobis distance between two bivariate distributions with different covariances
The question is pretty much contained in the title. What is the Mahalanobis distance for two distributions of different covariance matrices? What I have found till now assumes the same covariance for ...
7
votes
1answer
232 views
Finding the distribution of a statistic
Studying for a test. Couldn't answer this one.
Let $X_{1,i},X_{2,i},X_{3,i}, i=1,\ldots,n$ be iid $\mathcal{N}(0,1)$ random variables. Define
$W_i = (X_{1,i} + X_{2,i}X_{3,i})/\sqrt{1 + ...
7
votes
1answer
128 views
How to check for bivariate Gaussianity without the use of regression?
What steps could be taken to check for bivariate Gaussianity without using regression based check? Can we somehow employ the use of definition of variogram measure for assessing spatial variability?
7
votes
0answers
176 views
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 ...
6
votes
2answers
258 views
Probability density function between -1 and 1?
I'm currently using Gaussian distribution as a mutation operator for my genetic algorithm. However, I only want to obtain values between -1 and 1. I also don't wish to truncate my Gaussian ...
6
votes
4answers
670 views
R - QQPlot: how to see whether data are normally distributed
I have plotted this after I did a Shapiro-Wilk normality test. The test showed that it is likely that the population is normally distributed. However, how to see this "behaviour" on this plot?
...
6
votes
3answers
547 views
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 ...
6
votes
5answers
3k views
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 ...
6
votes
2answers
1k views
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 ...
6
votes
2answers
158 views
Inference with Gaussian Random Variable
Let $X = N(0,\frac{1}{\alpha})$, $Y = 2X + 8 + N_{y}$, and $N_{y}$ be a noise $N_{y} = N(0,1)$. Then, $P(y|x) = \frac{1}{\sqrt{2\pi}}exp\{ -\frac{1}{2}(y - 2x - 8)^{2} \}$
and $P(x) = ...
