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

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2
votes
2answers
73 views

If X is log-normal, is: $\frac{a}{\sqrt{b+cX}}$?

I am working for the first time with log-normal distributions and I want to verify whether the following statement is true. I am not sure whether all the properties of the log-normal distribution hold ...
1
vote
1answer
230 views

Probability - Normal Distribution, Heights of Women versus Men

The heights of young women aged $20$ to $29$ follow approximately the $\mathcal{N}(64, 2.7)$ distribution. Young men the same age have heights distributed as $\mathcal{N}(69.3, 2.8)$. Height is ...
0
votes
1answer
36 views

Minimum matching convolution (part II)

We assume we are working in $\mathcal{H}(\mathbb{R}^n)$, the space of real symmetric matrices. We define the partial order $\ge$ defined as $\Sigma_1\ge \Sigma_2$ iff $\Sigma_1-\Sigma_2$ is in ...
0
votes
1answer
33 views

Find $P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$ with a sample of $n=16$ and $X \sim N(50,100)$

If $X_1,X_2, ..., X_{16}$ is a random sample of size $n=16$ from the Normal Distribution $N(50,100)$, determine: $$P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$$ Okay well I know that ...
0
votes
0answers
20 views

how can we generate random numbers using skew normal distribution

I want to generate random numbers with skew normal distribution using rsn(). I can find the answer from the following link. how can we generate random numbers using skew normal distribution in ...
2
votes
1answer
167 views

Projection of Gaussian distribution along a vector.

Can anyone help me understand how to compute the projection of a 2D gaussian distribution along a vector. I intuitively realize that the projection will result in a 1D Gaussian, but I want to be sure. ...
0
votes
1answer
61 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
0
votes
0answers
55 views

Multivariate gaussian and average covariance matrix

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...
1
vote
0answers
25 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
2
votes
1answer
40 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
0
votes
0answers
80 views

Quadrant probability of non-centric bivariate normal distribution

Suppose $(X,Y)$ has a bivariate normal distribuion with non-zero mean vector $\mu$ and covariance matrix $\Sigma$. What should $\mathbb{P}(X>0,Y>0)$ be? My attempt gives me an definite ...
0
votes
1answer
64 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
0
votes
0answers
32 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
-1
votes
1answer
36 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
6
votes
1answer
105 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
0
votes
1answer
22 views

If X + Y is truncated normal and X and Y are identitically (but not independently) distributed? What is the distribution of X and Y?

Let $(aX + bY)$ be a truncated normal and assume $X,Y$ are both identically distributed (but necessarily NOT independent) what is the distribution of $X$ and $Y$? More importantly can the pdf of $X$ ...
0
votes
1answer
96 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
1
vote
0answers
54 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 ...
1
vote
0answers
78 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 ...
-1
votes
1answer
193 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 ...
2
votes
0answers
60 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
40 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
87 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
votes
2answers
70 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
vote
0answers
29 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
152 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
58 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
589 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
vote
2answers
93 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
37 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
56 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
51 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
40 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
55 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
26 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
27 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
68 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
vote
0answers
43 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
50 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
105 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
32 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
37 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
votes
0answers
26 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
votes
0answers
46 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
votes
1answer
59 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
41 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
19 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
51 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 ...
1
vote
1answer
30 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
29 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 ...