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

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2
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0answers
42 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 ...
1
vote
0answers
24 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
vote
1answer
34 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 ...
0
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1answer
32 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. ...
1
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0answers
19 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 ...
3
votes
1answer
150 views

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) ...
0
votes
1answer
14 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
votes
2answers
287 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
1
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2answers
31 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
votes
1answer
23 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
votes
0answers
52 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 ...
0
votes
1answer
37 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 ...
0
votes
1answer
23 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 $P(a < b)$? ...
2
votes
0answers
26 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
votes
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
votes
1answer
13 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) ...
1
vote
2answers
27 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}) = ...
1
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0answers
25 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 ...
0
votes
1answer
41 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
vote
2answers
35 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
votes
1answer
25 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
votes
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 ($A$) ...
0
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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
43 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
vote
1answer
34 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} < ...
0
votes
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
votes
1answer
42 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 ...
0
votes
0answers
37 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
vote
1answer
23 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 ...
0
votes
1answer
23 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 ...
0
votes
0answers
29 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 ...
2
votes
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} ...
0
votes
1answer
30 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 ...
0
votes
0answers
52 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 ...
1
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0answers
23 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 ...
1
vote
1answer
18 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 ...
0
votes
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.
0
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1answer
30 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 ...
0
votes
4answers
60 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)
1
vote
1answer
20 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 ...
2
votes
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 ...
0
votes
1answer
33 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
23 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 ...
0
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0answers
22 views

Normal Distribution and optimization

Suppose the radius $X$ (in mm) of certain kind of water pipes follows the normal distribution $N(\mu,1)$. If the radius is less than 10 or larger than 12, then it is failed product. Suppose the ...
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1answer
17 views

Normal approximation to Binomial probability distribution

Where did this 0.5 come from? I understand we are using Z-score but in my calculations I basically omit the 0.5 to get a probability of .9616.
0
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2answers
51 views

Equation with normal distribution function

I was working on a task in probability, and got stuck at this: $ϕ(\frac{x-50}{4}) - ϕ(\frac{-x-50}{4}) = 0.6$ ($ϕ$ is the normal distribution function.) It's so simple, yet I don't know what to do ...
0
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1answer
36 views

normal distribution strange probability

Given the particular normal distribution specified below, what is the probability that a random observation falls within the specified range .004 greater and less than the average? original Lower ...
0
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1answer
26 views

How to calculate covariance of X and Y given joint probability

$X$ and $Y$ are dependent variables both normally distributed as $N(\mu-const, \sigma^2)$. I don't know what the joint distribution is. I know that when $const = 0$, then the joint probability ...
1
vote
3answers
53 views

Normal distribution exercise!

If a technician does not encounters any hardware problems, the time he requires to assemble a computer follows a normal distribution with a mean of $30$ minutes and a standard deviation of $3$ ...