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

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10 views

Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
0
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0answers
33 views

Probability and continuous distributions

Suppose that the daily consumption of pepsi in ounces is normally distributed with normal(13, 4) in ounces. The daily amount consumed is independent of other days except adjacent days where the ...
0
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1answer
31 views

Questions about Variance and Covariance

I have a few questions about the linearity (or lack thereof) of covariance. Let $A_1, A_2.. An$ all be independent random variables that have the same mean $\mu$ and variance $\sigma^2$. (1) Would ...
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1answer
12 views

Distribution of test scores calculate cutoff given mean and standard deviation

A normal distribution of test scores has a mean of 38 and a standard deviation of 6. Everyone scoring at or above the 80th percentile gets placed in an advanced class. What is the cutoff score to get ...
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0answers
26 views

Help solving integration: $I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(a/\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}\right)dx$

My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$ $$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$ I ...
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0answers
19 views

Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ gaussian random variables with zero means and equal ...
1
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1answer
29 views

Gaussian distribution determined by first two moments

When said that Gaussian distribution is determined by it's mean and variance. How is that different of other distributions? Almost every distribution which I can think of has this property. For ...
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1answer
27 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
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0answers
17 views

Sum of two independent truncated gaussians

I'd like to ask for additional info regarding a previous post on the subject: Sum of two truncated gaussian but I can't comment directly on that. Assume $X \sim N(\mu_{1}, \sigma_1^2)$ is doubly ...
2
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0answers
47 views
+50

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
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1answer
12 views

How do I solve this question using Z Table and Normal distribution?

A company pays its employees an average wage of 15.90 an hour with a standard deviation of 1.50. Assume the wages are approximately normally distributed. a) what proportion of employees receive ...
5
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2answers
270 views

Triangular vs Normal distribution

I'm trying to approximate a standard normal distribution with a triangular distribution. What parameters of the triangular distribution (min, max and mode) are more suitable? Thank you
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2answers
30 views

Variance of |X-Y| for X and Y ~ N(0,1/2)

I know $X$ and $Y\sim\mathcal{N}(0,\frac12)$, $X$ and $Y$ are independent. I try the following way to solve variance of $g(X,Y)=|X-Y|$ ,which is $V(|X-Y|)$. If ...
0
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1answer
21 views

Type I error in Normal distributions

Let $X_1,\dots , X_n \stackrel{iid}{\sim} N(\mu, \sigma^2 = 4)$ Test $H_0: \mu = 10$ vs $H_1: \mu > 10$ take a random sample of $n=16$ and reject $H_0$ if $\bar{x}>14$ Find $\alpha$ the type I ...
2
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0answers
36 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
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1answer
34 views

Sum of two truncated normaly distributed variables

Let $X$ and $Y$ be two variables which are truncated normally distributed above zero (that is $X$ and $Y$ have the lower truncation point zero, their values are bounded above zero). Is $X+Y$ truncated ...
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1answer
22 views

Probability with intersecting normal distributions

There are two independent random variables $a$ and $b$, each distributed normally with their own parameters. Given the means and standard deviations for $a$ and $b$, how can I calculate $P(a < b)$? ...
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0answers
21 views

Normal Distribution Table urgent help [closed]

I have been asked to find this, $ P(−2 ≤ Z ≤ 2).$ I need to know how to find this, don't understand the last step when you add $0.5.$
2
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0answers
24 views

One-sided Bound on Sum of Fourth Moments

I'm interested in methods for proving one-sided bounds of the form $$ \mathbb{P}[\frac{1}{n}\sum_{i=1}^n X^4_i \geq 3+t]\leq Ce^{-nt} $$ where $X_i$ are standard normal random variables. I've run a ...
0
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1answer
19 views

Confidence Interval w/ true standard deviation?

I'm very scared that my calculations I did were wrong. Here is why: I assumed true standard deviation meant population S.D. However the question says the standard deviation is from a sample. So what ...
0
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1answer
11 views

Normal Distribution $r-1$ th moment with absolute value

I was stuck for this problem whole night and I tried numerical solution using MATLAB and the following result seems hold for x follow normal N(0,1) and for any positive number (not integer only) ...
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2answers
24 views

How to I use the standard normal table to get the following Z value?

So I am given that $P(X \le 31.5) = 0.05$ and according to the textbook, after standardizing and using the standard normal table we get $$(31.5 - \text{mean})/(\text{standard deviation}) = ...
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0answers
23 views

mean and variance of this Gaussian random variable

I am trying to read through this paper - http://www.malcolmdshuster.com/Pub_2002c_J_scale_scan.pdf Equation 2(b)from the paper says [A] $\nu_k \equiv 2(B_k - b).\epsilon_k - |\epsilon_k|^2 $ where ...
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1answer
40 views

Combining normal distrubutions

I am not sure of the terminology here, if this is a product, summation, or average. How can you take a two unimodal normal distributions and combine them into a bimodal distribution? And then combine ...
1
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2answers
34 views

Ratio of CDF to PDF increasing?

Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function. Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x? Numerically it seems to ...
3
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1answer
23 views

A tool weight is distributed normally with mean = $2265.4$. Given that 14% of the tools' weight are above 2278.36. what is the standard deviation?

A tool weight is distributed normally with mean = $2265.4$. Given that 14% of the tools' weight are above 2278.36. what is the standard deviation? Here the solution: denote $X$ as tool's ...
0
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0answers
30 views

Perturbed density of eigen-states of a 3 diagonal matrix

How does the density of eigen-states ($D(\lambda)$ is defined as $D(\lambda) d\lambda$ = Number of states in the range $\lambda ... \lambda + d\lambda$) of the following tridiagonal matrix ($A$) ...
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0answers
23 views

Distribution of Difference of Ordered Values Drawn From A Normal Distribution

This question has come up at least twice now when I was trying to estimate something*. I could always write out the integral or find it computationally but I'm hoping someone will give me an exact ...
0
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0answers
25 views

How to distribute a cost in a normal distribution

I need to spread out a number so that it reflects a normal distribution. For example, I have an item that cost $500,000$ dollars in year $2050$ and I would like to spread it across with a standard ...
0
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1answer
41 views

Normal distribution, $S^2$ distribution, and chi-square distribution exercise

Let $X_1,\dots , X_{16}$ be a random sample from a normal population with mean $\mu= 6$ and variance $\sigma^2 = 4$. (a) What is the approximate distribution of X? (b) Find $P( X< 4)$ (c) Find ...
1
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1answer
32 views

Normal Distribution Approximations and Central Limit Theorem

Let $X_1,\ldots,X_{144}$ be a random sample from a population with mean $\mu = 20$ and variance $\sigma^2 = 64$. (a) What is the approximate distribution of $\bar X$? (b) Find $P( \bar{X} < ...
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0answers
13 views

Relation between camera megapixels and signal to noise ratio

Disclaimer: I understand that this thing does almost nothing to photography (as noise is not so important to photography is self and because there are a lot of things influent to signal to noise like ...
0
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1answer
39 views

Calculate multivariate Gaussian from univariate Gaussian

I am currently trying to solve an exercise that involves estimating the position $\chi_t$ and and velocity $\dot\chi_t$ of a truck at time $t$. The truck moves on rails and is buffeted around by a ...
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0answers
34 views

For any $p$ we have $f\in D(\mathbb{R})$

Let $\varphi\in D(\mathbb{R})$ and $f=\varphi+p$ , $p$ is polynome $\star$ For any $p$ we have $f\in D(\mathbb{R})$ ?
1
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1answer
22 views

Normal Distribution Quartiles

lets say the first quartile of a random variable (continuous one) has a CDF function F such that $F(x) = .25$... e.g. the random variable is $3\times$ as likely to be larger than the first quartile ...
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1answer
22 views

Normalizing relative list of probabilities

I have an array of objects, and I want to randomly select one. These objects all have a performance property that ranges between [0, 1]. If this performance value is greater than (or equal to) some ...
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0answers
27 views

Converting normally distributed numbers to uniform distribution

I have a Perlin noise algorithm I've written my self. It seems to produce gausian numbers at the range of -1.5 and 1.5 but I'll convert them to the range of -1 and 1. I' currently working on a project ...
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0answers
43 views

Uniform convergence of functions involving normal CDF

Consider two sequences of continuous functions $(f_n)$ and $(g_n)$ for $n \geq 0$ defined by $$ f_n (x) := \int_0 ^t \Phi\left(\frac{x\Phi ^{-1}(\alpha(s) + \beta_n(s))+\Phi^{-1} ...
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1answer
29 views

What is the distribution of the sum of several normally distributed random variables?

Let's say we have n normally distributed random variables all with the same median and variance. Do we have a possibility to estimate the distribution law of the sum of those variables? I assume ...
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0answers
26 views

Normal distribution with unknown mean and variance

I've just done 3 pages of algebra having a go at this before realising I made a stupid mistake right at the beginning so it's all wrong. Moreover, I'm not sure my method was right (see below). I have ...
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0answers
42 views

Mean, variance and normal distribution

In a game of bridge hands of size 13 are dealt to each of 4 players in such a way that each hand can be considered to be a random sample without replacement from a standard pack of 52 cards. Each ...
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0answers
21 views

Coupling a chi-square to a normal random variable

Let $Z\sim \chi^2(k)$ be a random variable sampled from the Chi-Squared distribution with $k$ degrees of freedom. Vague question: Conditional on the value of $Z$, how can I reconstruct a sequence of ...
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1answer
17 views

Confidence ellipse for a 2D gaussian

For a 1D gaussian, the interval +/- 1SD about the mean will comprise ~68% of the area under the curve. Consider a 2D gaussian with a mean of zero and a diagonal covariance matrix (i.e., it is not ...
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1answer
11 views

The value of z representing the first Quartile of the standard normal distribution is:

I'm in desperate need of a hint at how they got the answer.
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1answer
29 views

Normal Approximation to the Binomial (Multiple Choice Question)

My first instinct in this question is use Normal approximation because N is large, and P is exactly between 1 and 0. I used the normal approximation, calculated when $p(X\le 19)$ and got 0.8997. The ...
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4answers
59 views

If $X \sim N(\mu, \sigma ^2)$, show that $(X - \mu) / \sigma \sim N(0,1)$ [closed]

I don't know how to do this. Do I need to use converge in distribution? (I thought this can only been used if $n$ involves)
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1answer
19 views

Argument shift Normal Distribution

In a mathematic book I have read following exercise: We throw a normal coin 10,000 times. The random variable $X$ tells us the number of tails. Give an approximation for $\mathbb{P}(4900 \leq X ...
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0answers
42 views

what is the expectation of $\sqrt{\left | x \right |} * sign(x)$ and $log(|x|)$ for a normal distribution

(1) What would $\int_{-\infty }^{\infty} \frac{\sqrt{\left | x \right |} * sign(x)}{\sqrt{2\pi}\sigma}e^{-0.5*\left ( \frac{x-\mu}{\sigma} \right )^{2}}dx$ evaluate to? This is expectation of ...
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1answer
31 views

Standard normal distribution inequality

I want to know how to prove the following inequality that seems to be true numerically. Let $n(x)$ be the density of the standard normal, and $N(x)$ be the cdf of standard normal. Then, for $x\geq ...
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0answers
21 views

Converting one normal distribution to another

I have a long data similar to this; between 15 and 25 consider its mean as m and calculated standard deviation using this formula. I assumed the above data as a normally distributed data and can ...