Questions on the Gaussian, or normal probability distribution, and related topics.
1
vote
1answer
74 views
Expected value of independent random variables
So I have this problem where I need to find $E(4X+3Y-2Z^2-W^2+8)$ where $W,X,Y,Z$ are all standard normal and I'm kind of confused on how to find the expected value here. I thought to do it we just ...
1
vote
1answer
85 views
One over a Normal Distribution
If X is a normal distribution $N(0,\sigma^2)$ is $\frac{1}{X}$ any sort of "official" distribution or something that should just be computed?
In particular I'm looking to find the expectation ...
3
votes
1answer
93 views
Is there a closed-form expression for the integral of this product of gaussian functions?
Considering:
$$f(x) = \frac{1}{\sigma_x\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x}{\sigma_x})^2}$$
$$g_i(x) = \frac{1}{\sigma_i\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{a_i+b_ix}{\sigma_i})^2}$$
Is there a ...
0
votes
0answers
58 views
integral related to Gaussian random variable and Brownian Motion
This integral arises from some work I am doing related to Brownian motion.
The integral of interest is the following
$\int^{\infty}_{t=0}\int^{b}_{x=-\infty}\frac{1}{\sqrt{2\pi ...
0
votes
0answers
27 views
Number of 1 in a binary string
I have a binary string $A$ of length $l$. I know in the string $p$ fraction is 1. So number of 1 is $pl$. I take a string $B$ of length $m$ from $A$. With 99.99% confidence, I want to find the upper ...
0
votes
0answers
105 views
normal distribution family tight
If we have $N(\mu_{n},\sigma^2_{n})$ distributions. I want to show that this family is tight if and only if the sequences $(\mu_{n})$ and $(\sigma^2_{n})$ are bounded.
A sequence of probability ...
2
votes
1answer
125 views
Acceptance probability of Metropolis-Hastings
I am an IT guy writing my masters thesis on MCMC methods for use in predicting the outcome of football(soccer) matches. Right now I am trying to wrap my head around MCMC and Metropolis-Hastings in ...
0
votes
0answers
63 views
Graphical demonstration of complex normal distribution (complex gaussian distribution)
I am looking for a graphical demonstration of complex normal distribution example.
Can anybody tell me a source for it?
1
vote
1answer
55 views
Computing average on normal distribution problem
A school makes a test and its scores has average $22.8$ and standard deviation $4.8$. The distribution of scores is approximately normal.
a) Choose one person randomly. What is the probability that ...
1
vote
4answers
252 views
The sum of n independent normal random variables.
How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean ...
0
votes
2answers
59 views
What is the equation of the expectation of the product of two normally distributed multivariate random variables?
Given two multivariate random variables $\mathbf{x} \sim N(\hat{\mathbf{x}}, Q)$, $\mathbf{y}\sim N(\hat{\mathbf{y}}, R)$ which are not independent but are correlated with covariance matrix $C$ and a ...
1
vote
2answers
179 views
Does a Chi-Square random variable $\chi^2_1$ mean that only one normal random variable was taken?
I'm trying to understand how Chi-Square variables work.
So far, I know that a Chi-Square random variable, $\chi^2$, means that one random value has been taken from a normally distributed graph. Let's ...
2
votes
2answers
318 views
Probability that the sum of all values of 5 pairs of dice will be between 30 and 40
I'm trying to solve a question that asks:
If 5 pairs of fair dice are rolled, approximate the probability that the sum
of the values obtained is between 30 and 40 inclusive.
My approach so ...
3
votes
2answers
351 views
Probability of one normdist being greater than another
I have two independant normally distributed random variables.
X ~ N(657, 3)
Y ~ N(661, 2)
P(x > y) = ?
How do I calculate the probability of X being greater ...
1
vote
2answers
177 views
Normal Distribution Question
Could someone go over these calculations and tell me where I'm going wrong please. It's to do with normal distribution.
The question: In a factory, the packets of sweets produced are supposed to ...
0
votes
1answer
39 views
Trouble with samples in a normal distribution
I'm okay with solving regular normal distribution questions (where X is a normal random variable with mean $\mu$ and standard deviation $\sigma$). However, we're currently dealing with samples within ...
0
votes
1answer
44 views
Quadratic functon integration over normal distribution
How one can prove that
$$
\int_{\mathbb{R}^N} \mathcal{N}(\mathbf{y}| \boldsymbol{\mu}_1, K_1) \log \mathcal{N}(\mathbf{y}| \boldsymbol{\mu}_0, K_0) d \mathbf{y} = -\frac12 \left[ N \log 2 \pi + \log ...
-1
votes
2answers
71 views
How to split $100$% to various components with different priorities?
For the following scenario :
$E=100$ (energy available = $100$%)
Components = $N$ (can be $1,2,3,\dots,\infty$)
Now I want to split $E=100$ for each component, however, components are prioritised :
...
0
votes
2answers
53 views
Distribution of two joint normal random variables
If two random variable $u_1$ and $u_2$ have a joint normal distribution what will be the distribution of the random variable $u_1-u_2$?
0
votes
1answer
189 views
Cumulative distribution function determine the random variable
I don't know that determine is the right word, but I try to explain. What I need to understand. :) So..
We know's that if a function fit this conditions:
Monotonically non-decreasing for each of its ...
0
votes
2answers
123 views
Fitting normal distribution to the data
I have been given a set of data points. How can I find the best fit of the form $$\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{\sigma^2}}?$$
Even better if Sage can do it. And how can I approximate ...
0
votes
0answers
104 views
Product of Two Multivariate Gaussian Distributions
I asked a question with a similar title before, but I phrased it incorrectly so I am reposting.
What is the mean and covariance of the distribution, $f_{PA}(PA) \cdot f_{Y|X,PA}(Y)$ where $f_{PA}$ ...
2
votes
1answer
160 views
Average euclidean distance between M normally distributed points
The result for the average distance between 2 points with normally distributed coordinates has already been demonstrated on this site and I found a white paper for the generalized result when these 2 ...
1
vote
0answers
61 views
Product of 2 Gaussian Distributions with Different Variables
Sorry, I asked the original question improperly so I am rephrasing it.
What is the mean and covariance of the distribution, $f_{PA}(PA) \cdot f_{Y|X,PA}(Y)$ where $f_{PA}$ and $f_{Y|X,PA}$ are both ...
1
vote
1answer
43 views
Scaling of a multivariate normal
We know that if a variable $X$ is iid from a $N(\mu,\sigma^2)$,
the distribution of $X+b$ is $N(\mu+b,\sigma^2)$
If we scale the $X$ by a scaling factor $k$, the new distribution will be ...
1
vote
1answer
49 views
What does it mean for a data set to have Gaussian-distributed noise?
I need to find an answer for this question.
What does it mean for a data set to have Gaussian-distributed noise?
Can anyone help?
1
vote
1answer
59 views
Gaussian distribution in Astronomy
What does it mean for a data set to have gaussian distributed noise?
What does an error bar on a data point really imply?
What does a 5-sigma result mean? How often is a 5-sigma result wrong?
What ...
0
votes
2answers
238 views
Statistics: normal distribution, finding truncated mean.
Say $X$ is a random variable arising from a normal distribution with mean $10$ and variance $4$ $(N(10, 4))$ truncated at $X=6$. How do I find the truncated mean of the distrubution?
1
vote
1answer
216 views
Distribution of integral of a normally distributed random variable
What can we say about distribution of
$\int_t^TN(\mu(s),\sigma^2(s))ds$
,where $N(\mu,\sigma)$ is independent normally distributed with mean $\mu(s)$ and variance $\sigma^2(s)$, $T$ and $t$ are ...
2
votes
1answer
43 views
Parameter optimization in probabilistic models
Task: Suppose we model a variable $y = Wx + \mu$ as a linear transformation of $x$ plus some Gaussian noise $\mu\sim\mathcal N(0,\sigma I)$. Our aim is to minimize the estimation error of $x$ given ...
0
votes
0answers
24 views
Where does the Gaussian function come from? [duplicate]
Possible Duplicate:
Where does the guassian function/normal or bell curve come from?
I have read countless pages on google and cannot find a satisfactory answer. I have also read ...
2
votes
4answers
279 views
Where does the guassian function/normal or bell curve come from?
I am confused as to where the function for the normal distribtuion comes from. Where does the e and pi come from? In my textbook I am presented with the function,but I am unsure about where it came ...
2
votes
2answers
132 views
Normal Distribution, The “Y” Value
Guys I am having trouble with the standard normal distribution.
http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm
We know the X values run from approx -inf to +inf but what are ...
1
vote
1answer
52 views
Determining most likely Gaussian distribution
I have two Gaussian distributions with mean and variance $(\mu_1,\sigma^2_1$) and $(\mu_2,\sigma^2_2)$. I then receive a series of values $x_1, x_2,...,x_n$ with mean and variance ...
0
votes
1answer
51 views
Calculating P(A>B), where A and B are normal distribution
In the problem we have that A ~ N(7, 11/60) and B ~ N(7.3, 7/20) and the question is what is the probability that A gives a higher value that B. Since the textbook we have for the course doesn't ...
0
votes
0answers
250 views
Linear combination of Normal Distribution
Suppose X is a N(0, 1) and Y is a N(1, 2). They have a cov[x, y] = 2. What is the distribution of (2*X - 3*Y).
I was thinking since they are not independent, any linear combination of them would not ...
2
votes
1answer
447 views
Approximate probability mass function into normal distribution
I have an array of values (mass function) Px and I want to approximate a normal distribution function (in Matlab) from them.
I can plot the mass function using bar(Px) and I would like to plot normal ...
0
votes
1answer
149 views
Normal Distribution problem; demand and inventory
This question is asked on an upcoming homework assignment:
The weekly demand for a product approximately has a normal distribution with mean 1,000 and standard deviation 200. The current on hand ...
1
vote
1answer
71 views
Variable t times a Wiener Process W(1/t)
If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that
Since $W(1/t)\space \sim N(0,1/t)$
that $V(t) \sim t\cdot N(0,1/t)$?
And if so then is $t\cdot N(0,1/t) = ...
2
votes
1answer
412 views
Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent
If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent?
What happens with the ...
1
vote
0answers
25 views
What is the product of the average x and the average (1/x), where x is normally distributed.
Has anyone ever seen a solution for the following...?
$$
\left( \frac{1}{n}\sum_{i=1}^{n} x_{i} \right) \times \left( \frac{1}{n} \sum_{i=1}^{n} \frac{1}{x_{i}}\right)
$$
where the x values are from ...
2
votes
1answer
149 views
On solving standard normal deviation equation numerically
Let $\mu$, $\sigma$, and $a$ be given real numbers and $\epsilon>0$ given. What kind of methods there are to solve the equation $$\dfrac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^x ...
0
votes
0answers
46 views
How to simplify this equation of matrix computation?
I need to simplify
$F=\Theta'_{A|B}\Delta_{A|B}\Theta_{A|B} - \mu'_A\Delta_{A|B}\Theta_{A|B}$
where
$\Theta_{A|B}=\Theta_A+\Delta_{AB}\Delta^{-1}_{BB}(\mu_B-\Theta_B)$
...
3
votes
1answer
58 views
Given that $X$ is normal, find the probability that $(X-10)^2 <12$
Suppose that $X$ is a random variable that has a normal distribution with mean = 5 and standard deviation = 10. Evaluate the following probabilities:
$\mathrm{Prob}((X-10)^2 < 12)$
0
votes
1answer
70 views
Normal Distribution Table
I have a pretty solid understanding of what a normal variable is but I am having difficulty understanding the graph. For example, what will the normal distribution probability be for these:
...
0
votes
0answers
51 views
Is normal distributed data more robust than skewed distributed one ?
I am learning about the skewed and normal distributions so I apology for some novice questions in advance. I know the definitions of them but I do not know if they can also indicate the robustness of ...
1
vote
1answer
596 views
Linear transformation of normal distribution
Not sure if "linear transformation" is the correct terminology, but...
Let $X$ be a random variable with a normal distribution $f(x)$ with mean $\mu_{X}$ and standard deviation $\sigma_{X}$:
$$f(x) = ...
0
votes
1answer
46 views
Is there a way to directly compute maximum of a sum of several Gaussian functions?
I have a problem which goes as follows. I am trying to predict the value of a variable $x$. I also have a set of measurements (the actual context is an image) $x^i$. I know from some training ...
1
vote
1answer
158 views
CDF of standard normal random variable never actually is 0 or 1, right?
The CDF of a standard normal random variable is never actually 0 or 1, they only approach 0 and 1, correct?
4
votes
1answer
99 views
What quality of a distribution describes the “spikiness” of its density, and how do I get a good density plot of a spiky distribution?
I'm a programmer, not a math guy, so please answer in English. ;)
Suppose I have a multi-modal univariate distribution like:
.. . .. ........... .. . .. .
...
