Tagged Questions

73 views

Independent gamma and normal distribution

Let $Z = X + Y$ where $X \sim N\left(\mu, \sigma^2 \right)$ and $Y \sim \Gamma\left(k, \theta \right)$ using this parametrization of the Gamma distribution. Also assume $X$ and $Y$ are independent. ...
84 views

Bivariate Normal distribution and correlation

Is the CDF of a bivariate normal distribution with mean $(0,0)$ and $\Sigma = ((1,\rho),(\rho,1))$ monotone in the correlation coefficient $\rho$?
40 views

Distribution of Extreme Spread for n, sigma

Simple form provided by WHuber: What is the distribution of the diameter of n points in the plane drawn iid from a bivariate Normal distribution? (Diameter is the greatest distance among any pair of ...
39 views

Does the sum of several distributions become more central or approximated to Normal

As the classic CLT states Xs follow the same distribution, then the sum of Xs approximate to Normal distribution. But what about several Xs follows different distributions (maybe the same class but ...
34 views

How to estimate a pdf of x under the model of y = x+n, when the pdf of y and the pdf of n are given

I guess I come up with a classic question, but I failed to find any useful solutions by far. My question is about the following model $$y=x+n$$ where $x$ is a hidden random variable that cannot be ...
56 views

Standard error from correlation coefficient

Many studies only report the relationship between two variables (e.g. linear or logistic equation), $n$, and $r^2$. I want to use these reported statistics to reproduce this relationship with its ...
142 views

Difference between Gaussian Distribution and Cauchy Distribution

I have searched for the above topic but did not have an answer. Can someone please tell me the detailed difference between the multivariate Gaussian Distribution and multivariate Cauchy Distribution? ...
74 views

kurtosis, positive skewed and negative skewed for probability distribution

When discussing probability distribution, I always read something such as excess kurtosis, positive kurtosis, positive skewed and negative skewed. What exactly do these concepts indicate? In practical ...
178 views

Calculation of an “unconstrained” normal distribution (starting from a censored one)

Assume that two r.v. $W$ and $Y|W=w$ with (1) $W \sim \text{N}(\mu_w,\sigma_w^2)$ (iid) (2) $Y|W=w \sim \text{N}(w,\sigma_y^2)$ (iid) Further we only observe $Y$ if $Y$ is less then $W$, i.e., ...
83 views

Derivation of conditional distribution from other two distributions

$$Y|X=x \sim N(x,1)\\X\sim N(\mu,\sigma^2 )$$ What distribution does $X|Y=y$ follow? My initial startegy was to $f_{Y|X}f_X=f_{X,Y}$ and solve for $f_{X|Y}=f_{X,Y}/f_{Y}$ . Computing for $f_{X,Y}$, ...
33 views

Generic test for unimodality given sample

Are there any generic tests to validate if a given sample follows a unimodal distribution, like a Gaussian, Cauchy, Student's t or a chi-square?
22 views

Monte Carlo by time or by interval

Say I compute monte carlo output from input scenarios. Input are discrete time series. I choose time series as an example to make the problem more obvious - this could be also any curve. Computation ...
41 views

Distribution of norm of random matrix

I am curious to know whether central-limit theorem like considerations hold true for special functions like the norm of a matrix. Specifically, I'm interested in the Spectral norm of a matrix ...
56 views

Question: What is a quadravariate distribution? Motivation: I found a reference to quadravariate distributions in Priestley (1981) Spectral Analysis and Time Series on p325, but no definition (the ...
82 views

When is the distribution of product of two normal distributed variables near normal distribution?

It is clear the product of normal distributed variables is not normal distributed. For example, if $X \sim N( \mu_1,\sigma_1^2)$, $Y \sim N( \mu_2,\sigma_2^2)$, then $XY$ does not has the ...
44 views

Properties of a particular generalization of the Gaussian-Gamma distribution

Consider the distribution $$P(x, p | \alpha, \beta, \lambda, \mu, p_0) = \mathcal{Ga}(p | \alpha, \beta) \mathcal{N}(x | \mu, \frac{1}{\lambda p} + \frac{1}{p_0}),$$ that is, a generalization of the ...
58 views

What is the distribution of linear model parameters?

I am interested in testing if linear models are statistically different, so I would like to know what I can assume about the distribution of linear parameter models, like the slope for example. More ...
30 views

Finding the normal divisor of a random variable

I have an iid sample $x_1, ..., x_n$ from a random variable $X$ which itself is a convolution $X = Z + \mathcal{N}(0, \sigma^2)$. Neither distribution of $Z$ nor parameter $\sigma^2$ are known (we ...
56 views

Calculating the std dev of a 30 team league with each team having a 50% chance of winning

I just want to preface this with the fact it is not a home work question. I am also not a stats person so I am not sure even how to start with calculating this, what it is called or where to look. ...
46 views

Ratio of sum of Normal to sum of cubes of Normal

Please help me to find the limiting distribution (as $n \rightarrow \infty$) of the following: $$U_n = \frac{X_1 + X_2 + \ldots + X_n}{X_1^3 + X_2^3 + \ldots X_n^3},$$ where $X_i$ are iid $N(0,1)$.
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How do I identify the “Long Tail” portion of my distribution?

I have a number of series that would typically be described as normal skewed or Gamma distributed. For example, say I have a group of customers and have calculated their spend over a fixed length of ...
79 views

Simple homework question about normally distributed variables

The question states: Consider a set of random variables $X_i$, where $i=1,...n$. Each $X_i$ is normally distributed with mean $0$ and variance $1$, i.e. $X_i$ are $\mathcal N(0,1)$. What is ...
113 views

Use the Central Limit Theorem to calculate the approx probabilities of Gamma RVs?

If $X_i, i=1,...,$ are independent, identically distributed $\operatorname{N}(0,1)$ random variables, and $Y_i = X_i^2$ are independent $\operatorname{Gamma}(\frac{1}{2},\frac{1}{2})$ RVs, use the ...
204 views

The sample size applied to a non-normal distribution

I have a single variable that represents my population values (sample of data): ...
100 views

Determining sample size in a non-normaly distributed data

I want to take some samples from a database nearly ~150.000 records of a non-normally distributed values. From this link I obtained some common procedures to find a sample, influenced by The level of ...
258 views

Generate distribution based on descriptive statistics

There's a hidden variable that I'd like to approximate. I know several summary statistics: min, max, mean, median, standard deviation, n; and that it's approximately normal. I can obviously do a ...
63 views

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Generalization of multivariate normal distribution and classification

I am interested in a family of multivariate distributions that can be seen as a generalization of the multivariate normal distribution, insofar as they are defined by an expectation value $\vec \mu$ ...
186 views

Inequality for bivariate normal distribution

Let $X_1$ and $X_2$ be bivariate normal with mean $\mu=(0,\mu_2)$, for any $\mu_2$, and correlation $\rho$. Consider the following inequality: \begin{align*} Pr\left\{|X_1| \ge ...
220 views

Order statistics of absolute value of bivariate normal distribution

Suppose $X_1$ and $X_2$ are bivariate normal and let $|X|_{(1)}$ and $|X|_{(2)}$ be the ordered version of their absolute value. I am interesting in finding the following probabilities or some bounds ...
228 views

Descriptive statistics for non normal data

When I use an histogram to plot my data in Stata almost all variables cluster around one value, i.e. these are either at 0 or 1 (a huge bar at one or 0). What does it show and how should I correct it? ...
2k views

How to determine whether data is slightly or extremely non-normally distributed?

I'm a PhD student and doing a research on regression analysis. My question is how to determine whether the data is slightly, moderately or extremely non-normally distributed? TQ to all responses ...
158 views

What are the connections between: the normal, the $\chi^2$ & the F distributions?

I have often read that there is a huge connection between the normal distribution and several other distributions. But these were only mathematical explanations. What's the "real" connection between ...
109 views

Continued fraction representation of the multivariate normal distribution [closed]

Can anybody help me with a continued fraction representation of the multivariate normal distribution? Such a representation is well-known for the univariate case; see, for example, C-I. C. Lee. On ...
92 views

Is there a name for distributions induced by norms?

The log density of a multivariate Gaussian is proportional to $||x-\mu||_2^2$, i.e. the norm of the difference between $x$, the random variable, and $\mu$, the mean of the distribution (assuming a ...
330 views

Is there any test for a null hypothesis of non-normality?

I'm currently looking for a test having for null hypothesis that the sample does not come from observing a normally distributed random variable. In other words, I'd like to know if there's a test ...
387 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 ...