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

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24 views

Sampling Distribution of the Mean

I want to know if my reasoning is correct. Let's say I got two normal distributed variables: Variable "X": 5.4 (mean), 2.856 (variance) Variable "Y": 5.4 (mean), 5.062 (variance) Let's pick 16 ...
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66 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 ...
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27 views

How to find value from Gaussian distribution for given point, covariance matrix and expected value.

While reading one article I came across that one of the values (probability) I am supposed to calculate is equal to N(v, b + (h^T)(W^T), I). Where b,v,h are vectors, W is a matrix and I is the ...
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29 views

Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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28 views

Range of sum of Normal Distribution.

May be its silly question but I was just wondering is there any way to find out the absolute range of sum of values of Random normal distribution of N numbers with mu and sigma as mean and Std. Dev. ? ...
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21 views

What type of distribution can be used to describe a game with positive expected winnings?

I've come across something I'm not too sure about. Let's say we flip a coin, heads mean we lose 1 unit, tails means we win a 1 unit. This distribution of outcomes in this would be considered normal, ...
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25 views

Log-likelihood of the normal distribution.

On the attached picture I've highlighted the term which I do not agree with. Is it actually true ? In my calculations I get $$-n(\frac{1}{2}\log(\sqrt{2\pi})+\log(\sigma)),$$ instead. Thank you in ...
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55 views

Poisson process. Finding 5th and 95th centiles

I am an undergraduate student of Economics. Today I was trying to solve 1 exercise related to Poisson process that I found confusing and I would be very grateful for your help, as my Mathematics ...
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105 views

Ito formula for jump proccess

I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have $dS_r=S_{t^-}\mu+S_{t^-}\sigma dB_t +S_{t^-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq ...
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7 views

References to papers/books that uses a kernel to smooth a discrete distribution

Since a kernel, such as Gaussian, is often used to smooth out the distribution of discrete points in 1D, 2D or 3D, I believe there must be some study materials or research work that have used this, ...
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993 views

Outliers in a Normal Distribution

Im doing AP Stat. in High School level. Here is a question i am stumped on because i feel like it is maybe a threory or law or something that i just never learned. However it DOES ask to show my ...
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18 views

Comparing normal distributions using a two sample Kolmogorov-Smirnov test

I have used a two sample Kolmogorov-Smirnov test to compare the distributions of two sets of data. I know that the K-S test is a non parametric test, however the distributions of data I'm comparing ...
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214 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
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17 views

Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
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92 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
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22 views

Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
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35 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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59 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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209 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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40 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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117 views

Expectation of normal CDF with truncation

Suppose that $a$ and $T$ are given positive numbers. I would like to evaluate $$\begin{align*} \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right],\tag{1} \end{align*}$$ where ...
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18 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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319 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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22 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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130 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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21 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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55 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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21 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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35 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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45 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 ...
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87 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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39 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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34 views

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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312 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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74 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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29 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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18 views

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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36 views

Frechet differentiability, asymptotic normality

I try to prove the asymptotic normality from the Frechet differentiability. Consider $$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$ and ...
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20 views

Linear Gaussian system, covariance of the normalisation constant

If we have the following multivariate Gaussian distributions: $$p(x) = N(x|\mu_x,\Sigma_x)$$ $$p(y|x) = N(y|Ax + b, \Sigma_y)$$ Now how can you deduce p(y) ? p(y) is called the normalisation ...
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31 views

intergral of the product of 2 multivariate Gaussian distribution

Suppose there are the following relationships between $x,y,w$, $$\begin{align}p(x,y) &= N(\mu_1, \Sigma_1)\\ p(x\mid w) &= N(\mu_2,\Sigma_2)\end{align},$$ is it possible to compute $p(y\mid ...
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65 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
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23 views

estimate normal distribution parameters by $n$ largest samples

If I have the $n$ largest out of $m$ values of a sample from independent normal distributed random variables $\mathbb{X}_1,\dots,\mathbb{X}_m\sim\mathcal{N}(\mu,\sigma)$ with unknown parameters ...
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31 views

What is the probability the maximum sample value comes from one of two random distributions?

Let $X_1$ and $X_2$ be randomly distributed variables with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$. Samples of size of $n_1$ and $n_2$ are drawn from each ...
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23 views

proof of As ~ N(A$\mu$, A$\Sigma$A')

assume that s is a vector of states which is distributed according to a gaussian with mean $\mu$ and variance $\Sigma$. A is the state transition matrix How can I proof that As ~ N(A$\mu$, ...
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68 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
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126 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
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83 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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157 views

Multivariate Distribution & Bayes Rule

Suppose I have that an unknown vector, x, where x is drawn from the following distribution$ \bigl(\begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$ ~ $N\bigl(0, \bigl[\begin{matrix} \sigma^2_1 ...
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155 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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36 views

Finding the distribution under a new measure

Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where $Z_t$ are standard normal variables. Find the distribution of ln($S_1/S_0$) under the Q measure given that dQ/dP is ...