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

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1answer
39 views

Calculate the asymptotic dystribution

Let $X_1,...,X_n$ be an i.i.d. random sample from a continuous distribution with density given by: $f(x;\theta)=(\theta-x)\frac{2}{\theta^2}$ if $0<=x<=\theta$ and 0 otherwise. We have the ...
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0answers
22 views

Attempted calculation of the probability to win a game.

I'm playing the game "Pepper Panic" and the goal is to create two pepper panics. I noted down some ten results by the numbers I obtained ($0$ or $1$). I obtained a mean of $0.4$ and a standard ...
0
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1answer
61 views

How to find the mean variable of a normal distribution with a given probability and standard deviation?

We have a machine that produces µ g of pasta to be stored in their package, with a standard deviation of 20g. It follows a normal distribution. And we don't want it to produce more than the package's ...
0
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1answer
70 views

how to calculate the marginal distribution of probabilistic principal component analysis

In the book Pattern recognition and machine learning from Bishop equation 12.33 states: $\mathbf{x} = \mathbf{W} \mathbf{z} + \boldsymbol\mu + \boldsymbol\epsilon$ Here $\mathbf{z}$ has a normal ...
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0answers
50 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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1answer
18 views

Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
0
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1answer
32 views

How to eliminate coefficients from a sum

For given random values $$X_i \sim\mathcal{N}(0,1)$$ and $$\frac{X_i-\mu}{\sigma}=\tilde{X_i}\sim\mathcal{N}(\mu,\sigma),\,\mu\in\mathbb{R},\,\sigma>0$$ prove ...
1
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1answer
183 views

Solving integral related to second moment of normal distribution restricted to interval. CAS answer acceptable

Let $f(x)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$, the pdf of the 1-dimensional normal distribution. Is it possible to compute $\int_{-a}^a x^2 ...
0
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1answer
295 views

How to prove Gaussian integral in normal distribution can be scaled to a standard curve?

If I want to solve the gaussian integral for normal distribution problems I only need to scale it to a standard normal distribution curve and consult a table. I want to know why this is valid (the ...
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1answer
61 views

Conversion to standard normal

How can I convert a the pdf of a normal distribution that it N(t,1), but integrated from 0 to infinity, to the standard normal. I found that the former is equal to 1- ϕ(-t) but i cant figure how this ...
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2answers
191 views

What is a short hand for random variable X, Y independent?

In some of the textbook problems, it would say, suppose X and Y are zero mean, unit variance independent Gaussians... I would usually just write $$X,Y \sim \mathcal N(0,1)$$ and this is a nice short ...
2
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1answer
162 views

Is the joint distribution of two independent, normally distributed random variables also normal?

Say I have $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, also $X$ and $Y$ are independent, then is the joint distribution of $X$ and $Y$ multivariate normal? I.e., $$\begin{bmatrix} ...
1
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1answer
54 views

Show that two sums have the same distribution?

I have not been able to show that the following two stochastic variables have the same distribution. My question is as follows: Let $$ X_1, X_2,..., X_n $$ be independent and identically ...
0
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1answer
52 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
0
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1answer
27 views

Calculating conditional mean of 2 Normal

If $\theta$ is $N(\bar{\theta},\sigma^2_\theta)$, and $s=\theta+\epsilon$, where $\epsilon$ is $N(0, \sigma^2_\epsilon)$, how can I derive that ...
3
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3answers
153 views

Why do we use only 1/2 for continuity correction in case of approximating binomial random varable to a standard normal random variable?

I have read about continuity correction in case of approximating a binomial random variable to a standard normal variable. But in all the examples , they only use 1/2 as a continuity correction ...
2
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0answers
36 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
3
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1answer
61 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
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2answers
31 views

Does it matter here that random variables are jointly normally distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
0
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1answer
33 views

covariance matrix in bivariate distribution

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I ...
1
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1answer
676 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
1
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1answer
62 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
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0answers
37 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 ...
0
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1answer
286 views

Dirac Delta function and normal distribution

I understand the Dirac Delta is the limit of a normal distribution when the variance of the normal distribution tends to 0: $$ \delta(x) = \lim_{v\to 0}\frac{e^{-x^2/2v}}{\sqrt{2\pi v}} $$ Then what ...
1
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2answers
77 views

What is the expected value of a standard normal random variable given value is positive?

Am not sure if I'm wording this correctly. But say we take huge sample of standard normal random variables. Then we separate out positive values. What would be average of the positive values ? What ...
0
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1answer
22 views

Standardizing Normal Distribution

I was listening to Statistics lecture on Normal distribution and I could not understand that how P(X-mean)/S.D<=(x-mean/S.D) becomes \phi (x- mean/ SD) got solved by chain rule.
2
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1answer
89 views

Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population.

Recall that for a $N(\mu,\sigma^2)$ population $W=\frac{n-1}{\sigma^2}S^2\sim \chi^2(n-1)$. [a] Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population. ...
3
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1answer
159 views

Estimate large covariance matrix using few samples.

Let $\mathbf{x}$ be a random vector in $\Bbb{R}^n$, such that $\mathbf{x}\sim N(\bar{\mathbf{x}}, \Sigma)$. $N$ observations of $\mathbf{x}$ are available, say $\{\mathbf{x}_i, i=1,\ldots,N\}$. The ...
0
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1answer
36 views

Joint distribution of two gaussians, one of which is dependent on the other.

Suppose $x\sim N(\mu_x,\sigma_x^2)$ and we are given that $y\mid x \sim N(a+bx,\sigma^2)$, where $a$ and $b$ are some constants. It is a fact that the joint distribution of $x$ and $y$ is a bivariate ...
2
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1answer
135 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
1
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1answer
54 views

Updating in game with normal distribution

In a game from the following paper, it is stated that Player $i$ observes a private signal $x_i = \theta + \epsilon_i$. Each $\epsilon_i$ is independently normally distributed with mean $0$ and ...
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0answers
29 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. ? ...
1
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1answer
44 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
0
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1answer
478 views

Density of $Y=|X|$, where $X\sim N(0,\sigma^2)$

I am reviewing for an upcoming exam, and came across this question in the textbook. Can someone please help me with this question? Thanks. If $X\sim N(0,\sigma^2)$, then find the density of ...
1
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1answer
18 views

Normal distribution of juice

Quantity of juice in a pack of 1L is normaly distributed with average (mean) 950ml, and with standard deviation of 10ml. What is the probability that random pack of juice contains less then 945ml of ...
0
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1answer
106 views

Intuition behind Normal distribution forumula

In this formula $$ P(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{ - \frac{ \left( {x - \mu } \right)^2 }{2\sigma^2}} $$ why do we divide by square root 2 pi and after that multiply everything by e in ...
3
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2answers
103 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
0
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1answer
76 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
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1answer
754 views

Do we need to use continuity correction if we use CLT to do normal approximation

In a hotel, large number of cups and saucers are washed everyday. The number of cups that are broken each day while washing averages $2.1$. The number of saucers broken each day averages $1.6$, ...
0
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1answer
63 views

Standard Normal Distribution and CDF

I have a data set which consists of measured time in seconds. Secs= ${3000, 3857, 2400, 3323}.$ Mean $\mu =3145$. Standard deviation $\sigma=609.556$. I calculated the Standard Normal variable for ...
5
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1answer
247 views

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
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0answers
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, ...
3
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1answer
71 views

Boundedness of an integral of square function implying zero integral

Let $\alpha:\mathbb R\mapsto\mathbb R$ be the smooth function such that $$\int_{-\infty}^{\infty}[\alpha'(x)-x\alpha(x)]^2e^{-\frac{x^2}2}dx<\infty.$$ I wish to prove that ...
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0answers
28 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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1answer
64 views

Beginner Econometrics question about probabilities for a normal variable

$Y \sim N(\mu, \sigma^2)\implies (Y-\mu)/\sigma$ Prove that this has a Mean of $0$ and a Variance of $1$.
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1answer
25 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
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2answers
386 views

Proof that if $Z$ is standard normal, then Z^2 is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of ...
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0answers
56 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 ...
1
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1answer
111 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
1
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1answer
109 views

Finding the 99% of a normally distributed graph

The heights of adults are normally distributed with a mean of 187.5 cm and a standard deviation of 9.5 cm. A standard doorway is designed so that 99% of adults have a space of at least 17 cm over ...