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

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1answer
24 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 ...
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1answer
412 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 ...
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1answer
50 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
32 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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1answer
165 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 ...
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2answers
45 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
18 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. ...
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0answers
16 views

How does one scale a covariance matrix learned on de-meaned and scaled data?

I have a dataset on which I want to train a multivariate mixture of gaussians. One common thing to do is de-mean and scale the data such that each feature has zero mean and unit covariance. If I ...
3
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1answer
138 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 ...
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1answer
26 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 ...
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1answer
61 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
52 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
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. ? ...
1
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1answer
38 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} ...
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1answer
200 views

If X ~ N(0,σ^2), find the density of Y = |x|

If X ~ N(0,σ^2), find the density of Y = |x| Hi I am reviewing for an upcoming exam, and came across this question in the textbook. Can someone please help me with this question. Thanks
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1answer
17 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
78 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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1answer
73 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| ...
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1answer
55 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
396 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 $1.6$, ...
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0answers
17 views

Normal distribution around and extreme value

I'm trying to create an artificial dataset with users, items, and ratings given by the users to the items. Creating the dataset, I pick the average rating for every item randomly, and let the ratings ...
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1answer
53 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 ...
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1answer
243 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, ...
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0answers
17 views

finding sampling distribution from standard normal

Let $X_1$, $X_2$, $X_3$ be a random sample of size $3$ from a standard normal distribution. Find the distribution of $X_1^2 + X_2^2 + X_3^2$. If possible, find the sampling distribution of ...
3
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1answer
67 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
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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1answer
53 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
22 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 ...
0
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2answers
158 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 ...
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1answer
90 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 ...
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1answer
78 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 ...
2
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0answers
63 views

Finite discrete approximation to the normal distribution

I wish to derive a finite (that is, which has a finite support) discrete approximation to a normal distribution, with the following considerations: It should have exactly the same mean and variance ...
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2answers
101 views

What is the reason for this answer on this coin problem?

Question: How many ways are there to pick a collection of 15 coins from bags of pennies, nickels, dimes, and quarters? (Assume coins of the same denomination are indistinguishable.) I know the answer ...
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1answer
83 views

Calculation of the $n$-th moment of the normal distribution

I need some advice on a question, maybe someone can give me a hint. Let $X \in N(0,\sigma^2)$, show that $E[X^{2n+1}] = 0$ for $n = 0,1,2,\ldots$, and that $E[X^{2n}]= [(2n)!/2^nn!]\cdot ...
0
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1answer
44 views

Stuck during calculations to show that $\mathbb{E}e^{uX}=e^{u\mu + 0.5u^2\sigma^2}$ for normal random variable $X$

I am reading S. Shreve's introduction to Stochastic Calculus. Exercise 1.6. (p. 43) should be a simple exercise, but I don't know how to continue. Let $u$ be a fixed number in $\mathbb{R}$ and $X$ a ...
2
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2answers
160 views

Expected distance between two vectors that belong to two different Gaussian distributions

Let $X$, $Y$ be two random variables that follow the Gaussian distributions with mean vectors $\mu_x$, $\mu_y$, and covariance matrices $\Sigma_x$, $\Sigma_y$, respectively. The probability density ...
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4answers
58 views

when to use which z score equation?

in some exam past papers I have been doing I have come across the statistics equation z=(sample mean - mean)/standard deviation as well as the equation z = (sample mean - mean)/(standard ...
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0answers
113 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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2answers
290 views

Normal distribution in which 90% of samples are between 2.99 and 3.01; what is the standard deviation?

Steel rods are manufactured to be 3 inches in diameter but they are acceptable if they are inside the limit 2.99 inches and 3.01 inches. It is observed that 5% are rejected as oversized and 5% are ...
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2answers
88 views

Find distribution $Y=X^2$

X~N(0,1). Find distribution $Y=X^2$ Can someone help me? I have no idea how to do it. I could try to start like this: $F_Y(t)=P(X^2<t)=P(-\sqrt(t)<X<\sqrt{t})$
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2answers
55 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
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1answer
45 views

Normal Distribution Worded Problem

Standard deviation = 2.5 mL 98% of bottles must be between 998 mL and 1000mL Pr( 998 < x < 1000) = 0.98 This is a technology exam question, therefore to find the mean I used the method: ...
2
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0answers
413 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
0
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1answer
35 views

Standard deviation in normal distribution

A manufacturer uses a machine to make metal rods.The diameter of the rods follow a normal distribution with a mean of 1cm and a standard deviation of 0.02cm If the standard deviation of the diameters ...
0
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1answer
54 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
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0answers
8 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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3answers
93 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...