Questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution.

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Given a histogram, programatically, how do I find the normal distributions that comprise it?

I will be getting data in at around 100 frames per second, and I need to compute the normal distributions that comprise a set of 48 data points. The distributions can partially overlap, but will ...
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40 views

Normal Distribution and Probability on Excel

The size of fish in a lake follows a Normal Distribution with mean m = 1 lb 4 oz and standard deviation s = 3 oz . Fish that weigh less than 1 lb 9 oz must be released back into the lake. Bill ...
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53 views

Expected value of normal distributed variable

I need to calculate the expected value of a modified normal distributed variable but i'm struggling. So maybe someone can help me. Suppose we've got a normal distributed variable $X \sim ...
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34 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
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124 views

Cumulative distribution function of a degenerate multivariate normal distribution

Let $X\in\mathbb{R}^{n}$ be a multivariate normal variable with the mean vector $\mu$ and the covariance matrix $\Sigma$. It is well known that if the matrix $\Sigma$ is positive-definite the ...
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68 views

MLE of MVN($\mu, \Sigma$)

I'm trying to find MLE of MVN($\mu, \Sigma$), i.e $N_k(\mu, \Sigma)$ with random sample $X_i, 1\le i \le n$. It was easy to get $\widehat{\mu}= \bar{X}$ and $\hat{\Sigma} = \frac{1}{n} \sum_i (X_i - ...
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145 views

Concept of Probability in math first level

I am trying to teach myself the concepts of probability and I was wondering if this is correct. I am only 13 years old and did not learn this yet. I am just reading parts of a probability book to get ...
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74 views

How to calculate Standard deviation with mean 0 and Min and max value on x-axis is -1 and 1 respectively?

How to calculate Standard deviation with mean 0 and Min and max value on x-axis is -1 and 1 respectively? It is of-course a normalize distribution. I apologize in advance for stupid question.
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142 views

Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
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62 views

An exercise about Borel paradox

If $X$ and $Y$ are independent standard normals, what is the conditional distribution of $Y$ given that $Z=1$, where $Z=I(X=Y)$?
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25 views

“fractional” expectation of zero-mean normal distribution

I'm trying to calculate $E[X^{\frac{2}{3} } ] $ of a zero-mean normal distribution. Any help to solve $$ E[X^{\frac{2}{3} } ] = \int\limits_{-\infty}^{\infty} x^{\frac{2}{3} } \frac{1}{\sigma ...
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57 views

Conditional density based on 2 gaussian measurements

However intuitive, I don't understand the formulas for the conditional mean and variance from 2 gaussian measurements. I have not found anything relevant mainly because I don't think I'm searching ...
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83 views

Normal distribution, how to calculate $\mu$ and $\sigma$

How to calculate $\mu$ and $\sigma^2$ when it is known just that $P(X\le 49)=0.6915$ and $P(X>51)=0.2266$ ? Thank you very much!
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116 views

Bivariate normal distribution; rotation; diagonal covariance matrix

Let $Z\sim N(0,\Sigma)$ with $$ \Sigma=\begin{pmatrix}\sigma_1^2 & p\sigma_1\sigma_2\\p\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix} $$ whereat $\sigma_i^2=\text{var}(Z_i), ...
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25 views

Binomial distribution vs Normal distribution

It is often said that the normal distribution "approximates" the binomial distribution. What is the precise mathematical expression of this fact?
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26 views

Expectation of product of Normal CDFs w.r.t. a bivariate Normal distribution?

I am trying to figure out if there is a closed form expression for the following expectation: $\int\int \phi(\gamma_1)\phi(\gamma_2) \mathcal{N}(\gamma\big|\mu, \Sigma)d\gamma_1 d\gamma_2$ where ...
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17 views

Modeling Gaussian Error

Context I am designing a simulation of a robot receiving input from a sensor which has gaussian error. The robot will start from a known position and move at a constant speed; the sensor will ...
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44 views

When do normal distributions not occur?

I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. For example, suppose ...
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177 views

How to use the normal probability table in reverse

I'm just wondering if anyone could give me a bit of advice on this. This relates to CCEA's S1 exam questions. $Z \sim \text{N}(0, 1)$ Let's say $\phi(z) = 0.5015$ Find z. Here is an extract of the ...
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64 views

moment generating function of normal distribution

I know this question relates to the chi-squared distribution, but I think what the question wants me to do is somehow derive this distribution from the information given. I have a normally ...
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97 views

Bivariate normal distribution question

If I have $(X,Y)$ with joint density $f(x,y)$ and $A$ is an invertible $2\times 2$ matrix, then for the random vector $(W,V)$ defined by: $$ \begin{pmatrix} W\\ V \\ ...
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48 views

A practical question in statistics

A student leaves home at 8 a.m. every morning in order to arrive at the University at 9 a.m. He finds that over a long period he is late once in forty times. ($\frac{1}{40}$) He then tries leaving ...
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19 views

What is the minimum standard deviation for a normal PDF such that one tail is always larger than that of a second normal PDF (different means)?

Say I have two weighted normal distributions, $$ f_1(x) = \frac{a}{2 \sigma_1} e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}} $$ and $$ f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} $$ ...
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127 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
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23 views

Exponential deviation with two $x$ values

I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ...
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19 views

Finding the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$

Suppose $X=[X_{1},X_{2}]$ and $X$~$N_2(μ,Σ)$. I wish to find the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$. Since this is of a quadratic form I do not know a way of solving this. However I kind ...
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80 views

Is normalcdf() inclusive?

I was looking at these examples here: Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is ...
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47 views

How to simplify the computation of a special case of multivariate normal cdf

I am trying to compute a multivariate normal cdf where all but the last bounds of the integrals are symmetric: $$F(a, \sigma, m ) = ...
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18 views

Continuity Correction with replacement

An urn contains 2 white and 8 red marbles. A marble is drawn from the urn 100 times in succession with replacement. What is the probability of drawing more than 75 red marbles? My attempt: $n=100, ...
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29 views

Average minimum distance

Let $\mathbf{u} =\begin{bmatrix}u_1 & u_2 & \dots & u_N \end{bmatrix}^T$ and $\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \dots & v_N\end{bmatrix}^T$. All the elements of ...
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31 views

Integral of cumulative normal

Let $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$ Question: for what values of $a$, $b$ and for what choices of $f(x)$ would the following ...
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54 views

Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
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44 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
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33 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
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58 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
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68 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
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38 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
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50 views

The number of coin tosses needed if the proportion of heads is to lie within 0.05 of p with probability at least 0.9?

There's a question I'm not really sure if I did it right or even understand what its trying to say. There is a coin which produces heads with an unknown probability p. How many times should we throw ...
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69 views

Normal distribution percentile calculation

I'm working out the following problem and there is a part that I am not understanding clearly. The weight distribution of parcels sent is normal with mean value $12$ lbs and standard deviation ...
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29 views

Calculate a probability involving drawings from bivariate normal variables with Xi and Yi i.i.d

There's a question which has been troubling me along with my earlier post. To be honest, I'm not entirely sure on how to proceed. All I know is that if X~N(mu,sigma^2) then P(X < A) = P(Z< ...
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Conditional covariance in gaussian graphical models

I have a hypothesis, but I'm not sure if its true. The Wikipedia page states that if the covariance matrix is given by $$\Sigma=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]$$ ...
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29 views

Calculating number on normal distribution curve

Can someone please let me know if I have this question correct: ...
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40 views

Where are they getting this number from?

Here's the question that I'm having a problem with: ...
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176 views

The characteristic function of a multivariate normal distributed random variable

The characteristic function of a random variable $X$ is defined as $\hat{X}(\theta)=\mathbb{E}(e^{i\theta X})$. If $X$ is a normally distributed random variable with mean $\mu$ and standard deviation ...
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60 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
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130 views

Convergence of a sequence of Gaussian random vectors

Let $X_n$ be a sequence of Gaussian random vectors that converges in distribution to some random vector $X$. Is $X$ Gaussian? Partial solution If we can show that $\mu_n := E\left(X_n\right)$ and ...
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49 views

Statistics - Lost with this question

I'm having trouble doing this question because I don't know where to begin. Could someone walk me through this slowly so that I understand the thought process and how to approach questions like this? ...
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26 views

We said the data is normally distributed, based on the raw data or residual?

I have a confusing regarding the assumption test for the data, in some theory were said that there are three assumption of data as we called as "good" data: Independent Normally distributed ...
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The space of all normal covariances matrices

Let $\cal C$ be the space of all $k-$variate normal covariance matrices and $\cal M$ be the set of all $k\times k$ symmetric positive semi-definite matrices. As we know that if $k=1$ then ${\cal ...
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107 views

help with Borel Cantelli lemma

There is a sequence of random variables $X_1,X_2,...$ For each i $X_i$ ~ $Normal(0,1)$ Is $ \frac{X_n}{n} \rightarrow 0 $ almost surely? Is $ \frac{X_n}{lnn} \rightarrow 0 $ almost ...