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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Two Gaussian Likelihoods with Two Decision Boundaries , 0/1 loss function

we assume that Y := {1, 2}. Then our decision can be re-written as y ∗ = {1 if p(x|y = 1) > p(x|y = 2) , 2 otherwise} with a decision boundary at p(x|y = 1) = p(x|y = 2). How can we construct an ...
3
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1answer
22 views

Distribution of random sample of confidence intervals containing the true mean

Given 50 random samples, each of size 25, from a normal distribution with mean 20 and standard deviation 5. From each of the 50 samples, you can find a 90% confidence interval for the mean. Let Y be a ...
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0answers
35 views

Best way to determine normality of data

I believe there are many ways to determine if data is normal or not: histogram shape QQ plot skewness kurtosis shapiro test. Which of the above is the best way to determine if data is normal and ...
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0answers
18 views

Cannot intuitively grasp “Standard normal deviate”

I cannot intuitively grasp the meaning of "Standard normal deviates". I think It would help if you provided me with either/all of the following: (i) real life examples of their application, (ii) an ...
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0answers
20 views

How to calculate distribution of Y|X from distributions of X|Y and Y [duplicate]

I'm trying to solve the following homework problem: Let $X$ given $Y=y$ have a normal distribution with mean $y$ and variance one, and let the marginal distribution of $Y$ be normal with mean ...
3
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1answer
97 views

Example of two *correlated* normal variables whose sum is not normal

I am aware of some nice examples of pairs of correlated random variables which are marginally normal but not jointly normal. See this answer by Dilip Sarwate, and this one by Cardinal. I am also ...
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1answer
13 views

Bayesian calculation of parameter multiplying normal

I am interested in a model like: $y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is. I ...
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0answers
29 views

Independence of functions of Random variables [on hold]

I am working with two RV's X and Y X is distributed normally with mean=x and variance=x^2 Y is distributed uniform (0,1) I know this... ...
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1answer
392 views

Why should we use t errors instead of normal errors?

In this blog post by Andrew Gelman, there is the following passage: The Bayesian models of 50 years ago seem hopelessly simple (except, of course, for simple problems), and I expect the Bayesian ...
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0answers
61 views

Normal Distribution with mean and standard deviation

I'm trying to solve the following problem: Suppose at breast height, the diameter of trees of a particular type is normally distributed with mean=8.8 inches and standard deviation= 2.8 inches.What ...
2
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1answer
34 views

Calculating the expected value and variance of an estimator of a normal quantile

I don't quite understand how to use the estimator function and the variance function and plug in the sample mean. I expected that we would plug in the value $\bar X - 1.645s$ into $E(s)$ and $V(s)$. ...
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1answer
39 views

How to make a vertical line in R [closed]

curve(dnorm(x,90,10),from=60, to=120) I need to make some vertical line in the x line that in number 70 and 110. How should I do it?
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0answers
22 views

Find the Distribution of a Function of a Multivariate Normal

I'm having trouble with the following question. Suppose $X \sim N(\mu, \Sigma)$, where $\mu \in \mathbb{R}^k$. Let $Y= AX +b$, where $A$ is an $n \times k$ matrix of constants and $b$ is an $n$ ...
-1
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0answers
35 views

Clarification on Central Limit Theorem [duplicate]

I have this (some_variable, frequency) data. Initially when I plotted the top 10% of this list. I got below, graph (zipf's graph) - Now with this data, I ...
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0answers
18 views

Distribution of noninjective function of random variables

Let $(X,Y)$ is a bivariate normal random variable wit mean $(0,0)$ and covariance matrix $\Sigma.$ Suppose that $T:\mathbb R^2\mapsto\mathbb R.$ I wish to compute the distribution of $T(X,Y).$ How ...
3
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1answer
45 views

Convert Poisson distribution to normal distribution

I primarily have a computer science background but now I am trying to teach myself basic stats. I have some data which I think has a Poisson distribution I have two questions: Is this a Poisson ...
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1answer
19 views

Finding $p(\tilde{y}|x)$ given measurement model and error distribution

Given two measurements of a variable $x$: $\tilde{y_1}=x+e_1$ $\tilde{y_2}=x+e_2$ where $e_1,e_2$ are zero-mean random variables following a bivariate normal distribution, with a known joint ...
3
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2answers
155 views

What causes non-normality of the error term in OLS?

In data, what causes the error term to be non-normally distributed in regression? Along the same lines, what solutions are there for non-normal residuals? For example, is it caused solely by a ...
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1answer
59 views

Estimation based on observing sum of two variables

Let $X_1,\dots,X_n$ are i.i.d normal $N(\mu,\sigma^2).$ Suppose that we only observe $$ X_1+X_2,\dots,X_1+X_n,\dots,X_{n-1}+X_n, $$ i.e, $X_i+X_j$ for all $i<j.$ I wish to find the best estimator ...
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1answer
30 views
+50

Help interpret Distribution of Wlan Signal Strength Measurements

For my project I need to evaluate large amounts of wlan signal strength measurements. Measurement is in dBm which is a logarithmic scale for milli watt (so every 3dBm the milliwatts double) where ...
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0answers
26 views

Distance between two independent normal random variable

What is the PDF of $Z=\sqrt{(X-x_0)^2+(Y-y_0)^2}$ when X and y are i.i.d. zero mean normal random variable (i.e., $x\sim N(0,\sigma^2)$ and $x\sim N(0,\sigma^2)$)
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1answer
30 views

Finding the probability that a set of measurement belongs to a set of normal distributions

Suppose that we are given a set of $n \cdot k $ normal distributions so that a given measurement of $k$ values either comes from a given set of $k$ normal distributions or not. How do we calculate the ...
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0answers
31 views

Estimating of variance of dependent normal distribution

Let $X_1,\dots,X_n$ are independent and identically unobservable variable on $\Omega$. Suppose that $f:\Omega\times\Omega\mapsto \mathbb R$ be unknown function such that we know the value of ...
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1answer
28 views

Most probable value given observation

Suppose I have observed $Z = 3$, where $Z = X + Y$, where $X \sim N(0,9), Y \sim N(0,4)$. How would I find the most probable value of $X$ that would have given me $Z = 3$? My attempt at a solution: ...
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0answers
29 views

Quad precision normal cdf and quantile functions

I'm looking to run the normal distribution cumulative distribution function and quantile function (its inverse) using the quadruple precision floating point format. Does anyone know of a library that ...
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1answer
48 views

Constrained MLE of multivariate normal

this might be obvious one but I have spent much time without gaining anything. If $\underline{X}$~$N_p(\underline{\mu},\sigma^2 I)$, where $\mu$ is known to lie on the unit sphere ($\mu^T\mu$), show ...
3
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2answers
63 views

Correlation estimation on Half-Normal Distribution

Let $$(X,Y)\sim N\left(\begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}\right)$$ and let we are observing $(|X_1|,|Y_1|),\dots,(|X_n|,|Y_n|)$ independently. I wish ...
3
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1answer
90 views

What is the median of an equally weighted mixture of two Normal Distributions?

Suppose men's heights follow a normal distribution $X \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and women's heights follow a normal distribution $Y \sim \mathcal{N}(\mu_2,\sigma_2^2)$. How can I find the ...
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0answers
17 views

When do we “bootstrap” and then use “t-dist” versus just using “t-dist”?

I'm taking a Data Analysis class on Coursera, and we are learning about bootstrapping when you have a small sample size (>30). What I don't understand, is when do you bootstrap and then use the ...
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2answers
340 views

Simulating draws from a Uniform Distribution using draws from a Normal Distribution

I recently purchased a data science interview resource in which one of the probability questions was as follows: Given draws from a normal distribution with known parameters, how can you simulate ...
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1answer
56 views

R code to solve for probability of normal distributions?

I don't understand which R code I am supposed to be using to figure these problems out. I brief explanation of what the code is doing would be amazing. The problems below are two different ...
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1answer
21 views

Back transformation with power function

I have the following distribution, where each observation represents a metric. $Metric = \frac{NExplored Nodes \times NGenerated Nodes}{NRepeated Nodes}$ This metric is highly correlated (0.99637) ...
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0answers
30 views

How do I use R code to solve for probability of normal distributions? [duplicate]

I don't understand which R code I am supposed to be using to figure these problems out. I brief explanation of what the code is doing would be amazing. The problems below are two different ...
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0answers
18 views

Linear Combination of Random Normal Variables

In order to prove that the linear combination of two independent normal distributions(say Z=X+Y) is normal, i am using their MGFs to show that the linear combination also has a similar mgf. This works ...
2
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2answers
116 views

Mean value of truncated normal distribution

I have a bunch of data where each observation represents an error $\in [0,1]$ (computed error between a variable and it's ground truth). Extra info: These are the results of the difference between a ...
2
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1answer
20 views

Percentage Beyond a Given Value for Empirically Defined Distribution

It is my understanding that standard deviation does not work well as a measurement for distributions that are heavily skewed. If I have a heavily right-skewed distribution, should I simply use the ...
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1answer
30 views

Why constrain mean and standard deviation when proving Gaussian is maximum differential entropy pdf?

I'm reading Bishop's Pattern Recognition and Machine Learning. In chapter 1.6: Information Theory (page 53) when trying to derive the maximum differential entropy pdf from the definition of continuous ...
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2answers
166 views

Why do we have to assume normality for a one-sample t-test?

As a consequence of the central limit theorem the sampling distribution of the sample means will always be normal whatever is the distribution of the variable we measure. From our sample we can ...
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1answer
32 views

degenerate univariate Gaussian

I was watching a video on Gaussian distributions and it defined the degenerate univariate Gaussian as a Gaussian where the variance is zero. However, I am really struggling to understand how the ...
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1answer
59 views

How do I use the Hessian matrix for maximum likelihood estimation?

I am trying to teach myself maximum likelihood estimation using the Newton-Raphson method and related iterative methods. I don't understand the link between the hessian, the expected value of the ...
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1answer
33 views

Variance and asymmetry on relative frequency class distribution

I don't know how to resolve this (easy) exercise. I've calculated the first output. But I don't know if it's correct. Calculate arithmetic mean, variance (standard deviation^2), concentration and ...
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0answers
34 views

Estimate the covariance matrix of a normal distribution if the mean vectors is given by a linear rule

Let $X=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ be a random vector that follows a multivariate Gaussian distribution with known mean vector $\mu=(\mu_1,\ldots,\mu_n)^\top\in\Bbb{R}^n$. The covariance matrix ...
2
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1answer
58 views

Analyzing the effect of categorical variables on a correlation coefficient

For my research project, I’m looking for some help on how to analyze my data. The research setup is as follows: I’ve got two normal variables that I want to correlate with each other and a number of ...
3
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2answers
109 views

Sampling from a product of two Multivariate Gaussians

I have a multivariate Gaussian defined as follows: $$ p(x) = \omega(x)\gamma(x) $$ where $\omega$ and $\gamma$ both are multivariate Gaussians and from which I can sample very efficiently given due to ...
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1answer
25 views

Percentage with Normal Distribution

Based on the picture below, there is a value with a percentage that is 68 and 95 percent in relation to standard deviations in a normal distribution. My question is: What do the 68 and 95 percentages ...
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0answers
15 views

Fitting data in multivariate Gaussian

I have a dataset of N*d feature vectors and I was asked to fit them in a multivariate gaussian, with a matlab function (that someone else has programed) that recieves the number of points, mean and ...
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2answers
50 views

Normality test for repeated measures data

I've measured the motor function of the same subjects (n=6) over 7 different time points. I would like to know whether the mean motor function varies significantly with time (day 1 versus day 2 etc) ...
2
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1answer
77 views

How to calculate variance of Gaussian given mean and average deviation

I want to calculate the variance sigma of a Gaussian (normal) distribution given its mean mu and average deviation d (i.e. average of absolute value of difference from the mean). How can I do this? ...
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2answers
77 views

Predictive posterior distribution with multivariate normal distribution

Suppose I have a multivariate normal ${\bf{Y}}|{\bf{\theta}} \sim {\bf{MVN}}(X {\bf{\beta}}, \sigma^{2}H(\phi))$ where ${\bf{Y}}$ is a set of observations ${\bf{Y}} = ...
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0answers
21 views

How to model the sampling distribution of the sample sum

I'm stuck on a stats problem and am wondering if someone can point out the error in my logic. Imagine you are planning for a camping trip for 50 people. Each person consumes 2.0 lb of food per day on ...