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

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667 views

conditional expectation of normal distribution using sigma algebra

Suppose $X$ and $I$ and independent, $X$ has a standard normal distribution and $I$ take values $1$ and $-1$ with equal probabilities. Let $Y = IX$. How would I find the distribution of $Y$ and ...
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2answers
97 views

Ranking students from 2 separate exams in single scale.

Is there a way to rank 2 student groups who face 2 separate exams in a single scale using z-score, given that there are enough student in each group to consider each score distribution a normal ...
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1answer
177 views

A simple question about normal distribution

Suppost we have a dataset as below: (Value,Frequency) pairs: (1,2), (2,4), (3,6), (4,8), (5,10) Can we say that this data is normally distributed, or have a normal distribution for this dataset?
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1answer
177 views

Standard normal distribution probabilities

Ok so I am having difficulty understand the concept behind standard normal distribution probabilities, in the questions I am getting a graph and a table FILLED with numbers, top header column has ...
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1answer
422 views

Average Value of Bounded Normal Distribution

Suppose a truck has a capacity of 100 and order sizes to be filled are normal distributed with mean 95 and standard deviation of 10. There is about 30% chance that capacity is exceeded. In this case ...
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1answer
223 views

Normalize only big numbers for plotting

I have a set of numbers: [9, 8, 6, 4000] I want to plot a bar chart and I want to normalize only the 4000 number to 4, so the range of Y axis will be [0, 9]. Under the 4 bar I would write * 1000 so ...
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1answer
38 views

Distribution of the lengths of a component question

I've been working through my notes on the normal distribution and I'm currently struggling with the whole of question i). Would appreciate any suggestions or advice on how to tackle it. There are ...
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1answer
862 views

How to generate noise signal?

What is the simplest formula of some noise signal? $A(t)=...$ where t is time. What is the name of a noise, which power spectral density is gaussian? EDIT 1 Actually I need a function which can ...
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1answer
522 views

Prove that vector has normal distribution

You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$. Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$. You build following random variables, based ...
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1answer
1k views

Chi-square approximation to standard normal (0,1)

Supose that $S_n$ has a $\chi^2$ distribution with $n$ degrees of freedom. Show that $$ \mathbb{P}(S_n \le x) = f\left(\sqrt{2x}-\sqrt{2n}\right) $$ where $f(u)$ is the normal distribution. I tried ...
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1answer
33 views

Scaling a range of values without a known maximum

I want to know is it possible to scale a set of numbers without knowning the upper limit. Say for example I have 1000 number values. I want to plot each of these values within a range of 0- 90. Is ...
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2answers
112 views

Finding a narrower confidence interval for a given CI, sample mean and size

I'm trying to understand confidence intervals but having some trouble. I've been doing some exercises I found online and I'm stuck on this question: I have been given a 95% confidence interval for a ...
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1answer
1k views

conversion of 2D Gaussian into polar coordinates

Is it possible to convert the 2D Gaussian function in to polar coordinates? ...
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0answers
13 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
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0answers
10 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes $||X||$ ...
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0answers
19 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically ...
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1answer
9 views

When the domain of a continuous distribution exceeds feasible values, what should I do?

Now I need a (maybe approximated) model for this distribution: $$X=(x_1, x_2, …, x_n)$$ where $x_i$ is a real number between $0.0$ and $1.0$, and the sum of $x_i$ equals $1.0$. Now, I want to use ...
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0answers
17 views

MLE of variance for a spherical Gaussian

I am trying to implement the X-Means clustering algorithm. In it, the authors use the BIC to determine which model fits the data best. It is explained here: ...
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0answers
24 views

How good of an approximation is a normal probability distribution for sum of dice rolls?

I want to know how well the normal distribution explains the sum of rolls with n dice with s sides. The mean value and the variance of the dice rolls are $$\mu=n\frac{s+1}{2}$$ and ...
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1answer
25 views

Approximation of distributions with dice

I want to know what dice to roll to get a given probability distribution(mainly normal distributions but exponential distribution would also be helpful). I want a function $f$ so that ...
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0answers
11 views

Does Bivariate Normal have an MLR?

In general, with all parameters unknown I think the answer to this question is no. I think this because in this instance we would have a curved multivariate exponential family. Is this reasoning ...
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0answers
39 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
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1answer
11 views

Scaling Normal Distribution

Why is it that $N(0, ct) = \sqrt c N(0,t)$? What does it mean when we take a constant out of a distribution?
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47 views

Find $E[X+2Y|Z]$

$X,Y$ are independent standard normal. Let $W=X+Y$, $Z=X-Y$. Find $E[X+2Y|Z]$ Attempt: $E[X+2Y|Z=z] = E[X+2Y|X-Y=z] = E[Y+z+2Y] = 3E[Y]+Z = Z$ Is this correct?
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1answer
15 views

Integrate bivariate normal distribution over circular region

Context: Need to compute the probability that a 2D Gaussian random walk falls within distance $ d $ of some point $ p $ on the next step. (Assume the covariance $ \Sigma $ is the identity matrix $ I ...
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23 views

distribution of quadratic form of jointly normal random variables?

I need to derive the distribution of the random variable $\frac{W'(I-1(1'1)^{-1}1')ZZ'(I-1(1'1)^{-1}1')W} {Z'(I-1(1'1)^{-1}1')Z}$ , where $(Z, W)'$ ~ $N(0, I), \,Z=(Z_{1}, ..., Z_{n}), \,W=(W_{1}, ...
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2answers
35 views

Histogram of random numbers from normal distribution

If I generate, say, 10000 numbers from the normal distribution (in Matlab) and want to draw a histogram with 10 bins, it resembles the normal distribution pretty accurately. However, if I decide to ...
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0answers
38 views

Exponent - Solving for an unknown within an expectation

I have reached a stage where I need to solve for an unknown number, $\theta$ . However, I stuck and don't know how to proceed further. The equation to be solved is: $E\left[ \exp(\theta a^i) * ...
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1answer
33 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
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24 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
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30 views

If B is a N(0,1) R.V., show $E[B^4] = 3$

I've read in Elementary Stochastic Processes by Mikosch (p. 98), that it is a well known fact that: If B is a N(0,1) R.V., $E[B^4] = 3$ I also see something equivalent (but uncited) on the ...
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17 views

Scaled distribution of Brownian motion

If I have $X = 5(B_t - B_s)$ Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ? Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$ Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z ...
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1answer
23 views

How to extract a covariance matrix with this information

Referring to the above image, I wanted to know how to get the covariance matrix $\sum$. My understanding is, $A$, is our transformation matrice, such that $\begin{bmatrix} X_1 \\ X_2 \\ ...
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3answers
39 views

Finding the probability of loss from standard deviation in normal distribution

I am unsure how to approach the following question. The returns from a project are normally distributed with a mean of \$220,000 and a standard deviation of \$160,000. If the project loses more than ...
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1answer
18 views

Understanding the normalization of a Gaussian

I have a Gaussian defined as follows: $W(\theta) = j * exp(-0.5 * \theta^2 / \sigma^2)$. I want to set $j$ such that $\frac{1}{360}\int_{-180}^{180} W(\theta)d\theta = 1$. I'm using two values for ...
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16 views

Normal distribution tables - right or left?

Are the probabilities in normal distribution tables given typically to the right or left of the $Z$ score? One such text I am reading says to the right. However, in my lecturer's exercises, I ...
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2answers
18 views

Calculating a normal distribution with a sample size?

the sample of $n=25$ is what is throwing me off. I have no clue what to do with it. Given a normal distribution with $\mu=101, \sigma=25$, and given you select of $n=25$ $A.)$ $P(\overline{X} ...
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24 views

Approximating the integral of the exponential of a quadratic function

An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and ...
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18 views

Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
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25 views

Can a mixture of normals be a constant?

Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?     ...
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45 views

If $X|Y$ and $Y$ are both normal, is $X|Y>y$ normal as well?

Consider two random variables, $X$ and $Y$, with the following properties: $X|Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X|Y>y$ follow a normal distribution as well? If so, what are its ...
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35 views

Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...
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1answer
29 views

Random Gaussian variable raised to arbitrary power

Given $x$ that follows a distribution $P(x)=e^{\frac{-x^2}{2\sigma^2}}$ i.e. a random Gaussian variable, can I say anything about the distribution of $x^n$ for fixed $n$? Specifically, is there ever ...
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0answers
15 views

sampling from a multivariate guassian: intuition behind using cholesky decomposition

I'm trying to understand how sampling from a multivariate gaussian works and why the cholesky decomposition is a way to do it. Let's say we have a 25 dimensional multivariate with a 25x25 covariance ...
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1answer
49 views

Smallest n to align sample mean with population mean

There's a question in my book that I just do not understand. This is it in its entirety: Let $ \bar{X} $ be the sample mean of a random sample of size $ n $ from a normal distribution with a variance ...
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34 views

How to obtain the pdf of this transformation of normal random variables?

Given the following: Let $r = ab + n$, where $a, b,$ and $n$ are independent zero-mean Gaussian random variables with variances $\sigma_a$, $\sigma_b$, and $\sigma_n$, respectively. Find the MAP ...
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19 views

Bivariate normal exercise - check my answer please

Similar to the question I asked before, with one subtle difference. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are correlated with $\rho = 0.5$ then find: $a)$ the covariance ...
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1answer
23 views

Bivariate normal exercise - check please

I am trying to self learn some probability and wanted to ensure I was getting these exercises correct. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are independent, then find: $a)$ ...
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21 views

Proving a process is a P Brownian Motion

Let $X_t = tW_{\frac{1}{t}} \forall t>0$ and $X_0 = 0$. I am trying to show that this process is a brownian motion under some measure P. I have shown that it is continuous and that it is ...
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2answers
72 views

Sum of a Normal and a Truncated Normal distribution

I have normal distribution $ N(\mu_1, \sigma_1)$ which shows the amount of demand in warehouse 1. The current amount of stock in the warehouse 1 is C. If the random demand is greater than C, it cannot ...