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

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0
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3answers
451 views

What does it mean for two random variables to have bivariate normal distribution?

The following is Sheldon Ross's definition: We say that the random variables $X,Y$ have a bivariate normal distribution if, for some constants $\mu_x,\mu_y,\sigma_x>0,\sigma_y>0, ...
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1answer
501 views

Conditional probability distribution with Gaussian noise

If I have a relationship as follows: $$Y = a X + G(0,\sigma^2),\text{ so }y = a X + \text{some Gaussian noise}.$$ The conditional probability distribution of $y$ given $x$, i.e. $P(y|x)$, is equal ...
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1answer
158 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
1k views

How to apply Central Limit Theorem to Uniform Distribution to generate Normal Distrubution?

Suppose I have a simple uniform continuous "unit" distribution X: $$\begin{align*} \forall y \in \mathbb{R} \implies \\ y < 0 : & P(X < y) = 0 \\ y \in [0,1] : & P(X < y) = y \\ ...
0
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1answer
209 views

Upper-bound on the sum of two dependent Gaussians.

Let $X$ and $Y$ be two dependent normally distributed continuous random variables (their marginals are $\mathcal{N}(0, 1)$). I would like to find an upper bound on the probability that one is greater ...
1
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1answer
361 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 ...
0
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1answer
270 views

Density of truncated normal distribution?

I have a truncated normal distribution with mode $0$ and variance $\sigma^2$ that only consists of non negative values. What is the density of this distribution at some non negative $x$? I have just ...
1
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1answer
196 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
32 views

Finding the Z value from the tail area of a distribution

Hi everyone. I'm really struggling with question v). I can't look up 0.99... Here are the answers I have so far: i) 1.64 ii) -1.64 iii) 2.05 iv) -2.12 Thanks ...
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1answer
37 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 ...
0
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0answers
77 views

Normal Distribution and Distributions

If $ X(w,t)=\xi +\eta t $, $\xi$ and $\eta$ are random variable which each of them has normal distribution with parameter $(a,\sigma^{2})$, then compute $P_{t}(X)$ and $K_{X}(t_{1},t_{2})$. Edit: ...
1
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3answers
176 views

Computing Some Integrals via Gauss Integral

$ \displaystyle\int_{-\infty }^\infty e^{-\frac{1}{2} x^2} \; dx $ and $ \displaystyle\int_{-\infty }^\infty x^{2}e^{-\frac{1}{2}x^2} \; dx $ how i compute these integrals via Gauss Integral?
3
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1answer
144 views

Estimating number drawn from one distribution based on sum of that number and number drawn from another distribution

I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve. A computer draws a number $x$ at ...
2
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1answer
3k views

Prove one standard deviation lies on inflection points

Is my conjecture correct that one standard deviation lies on the inflection points of the normal distribution curve (of the probability density function)? How can it be proved using the standard ...
1
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2answers
319 views

General characteristics of a normal distribution

If the normal distribution curve is symmetrical about the vertical line then the mean = mode = median This would mean that: ...
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1answer
445 views

Normal distribution involving $\Phi(z)$ and standard deviation

The random variable X has normal distribution with mean $\mu$ and standard deviation $\sigma$. $\mathbb{P}(X>31)=0.2743$ and $\mathbb{P}(X<39)=0.9192$. Find $\mu$ and $\sigma$.
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0answers
118 views

Incrementally compute the conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...
1
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1answer
146 views

identically distributed question

$X$ and $Y$ are independent normal random variables and have the same moment-generating function and are thus identically distributed. Find the distribution of $Z$ where $Z=aX+bY$. I have the ...
0
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1answer
251 views

Gaussian formula for $n$ dimensions

I know this is a very simple one. If this is the formula for the two dimensional Gaussian (no covariance matrix considered - I have one mean and variance for each dimension): $$ A\exp{\left[ ...
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0answers
125 views

How can I find Multivariate normal distribution original paper? And other nature-revealing articles by the way

I want to see how Gauss get this distribution function representation. I want to understand deeper of Multivariate normal distribution. I tried but failed to search the original paper of Gauss ...
2
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1answer
360 views

Combining 1D normal distributions into a 2D distribution

First of all, apologies for my poor terminology - I have a particular problem which I understand in own terms, but I am having difficulty in applying the mathematics in the correct manner. My problem ...
1
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0answers
342 views

Mean value from part of normal distribution

I have a problem to solve. Lets say that there is normal distribution with mean value 5000 and deviation 1000. I have to know lets say what is a mean o 25 percent biggest numbers. How to calculate ...
1
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1answer
689 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 ...
6
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1answer
730 views

Can the product of two non-independent Gaussians be Gaussian?

We recently discussed this: Is the product of two Gaussian random variables also a Gaussian? What was established was that in nontrivial cases (i.e., ruling out zero-variance Gaussians, which are ...
11
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3answers
19k views

Is the product of two Gaussian random variables also a Gaussian?

Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed? Is the answer any different if we know that $X$ and $Y$ are independent?
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1answer
411 views

Q-Test in mathematica

I want to use the Q-test to determine outlier in my dataset. However, I cannot find a function to do this is in Mathematica. Does anyone know if there is a function which does this in Mathematica? ...
5
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1answer
127 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma ...
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1answer
7k views

What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
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1answer
462 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 ...
6
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1answer
7k views

Calculation of the n-th central moment of the normal distribution $\mathcal{N}(\mu,\sigma^2)$

Since integration is not my strong suit I need some feedback on this, please: Let $Y$ be $\mathcal{N}(\mu,\sigma^2)$, the normal distrubution with parameters $\mu$ and $\sigma^2$. I know $\mu$ is the ...
1
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1answer
223 views

μ and σ calculations for a random variable with a normal distribution [duplicate]

Possible Duplicate: Calculating mu and sigma (μ and σ) of a normal random variable If I have a random variable X with parameters μ and σ unknown. It is known that $P (X \ge 75) = 0.7764$ ...
1
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1answer
814 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 ...
5
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1answer
3k views

X,Y are independent standard normal distributed then what is the distribution of $\frac{X}{X+Y}$

X, Y are independent standard normal random variables, what is the distribution of $$ \frac{X}{X+Y} $$ Could anyone help me with this? Thanks. I have worked the problem by multivariable ...
1
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0answers
82 views

Inequality about normal distribution [duplicate]

Possible Duplicate: Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF Let Z be a standard normal random varible. How to prove that: ...
0
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0answers
108 views

Application of bell curve

I'm not a mathematician, hence why I'm here. I would like some help with some references please so as I can prove a point. Now, in work we are being assessed on our quality of work using the bell ...
22
votes
2answers
7k views

Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? ...
2
votes
1answer
104 views

$N(p, p q/n)$-distributed

Let $Y$ be a $N\left(p, \frac{p q}{n}\right)$-distributed random variable. We want to minimize $$ \delta = \delta(n)$$ so that the possibility that $p$ covers the interval $$ J = [Y-\delta, ...
3
votes
1answer
4k views

Expected value of normal distribution given that distribution is positive

Given $X \sim N(0, \sigma^2)$ (that is, $X:\mathbb{R} \to \mathbb{R}$ is a normal random variable with mean $0$ and variance $\sigma^2$), I'm trying to calculate the expected value of $X$ given that ...
2
votes
3answers
8k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
1
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2answers
205 views

Integral over the full support of a square and cube of a convolution of normal and uniform

I've got a uniform random variable $X\sim\mathcal{U}(-a,a)$ and a normal random variable $Y\sim\mathcal{N}(0,\sigma^2)$. I am interested in their sum $Z=X+Y$. Using the convolution integral, one can ...
2
votes
0answers
204 views

Convergence rate for the p.d.f. of a normalized mean to Gaussian (i..e Berry-Esseen for pdfs)

Berry-Esseen Theorem states that the rate of convergence of the probability distribution of normalized sample mean converges to Gaussian at rate $O(1/\sqrt{n})$ (given that certain conditions are met, ...
0
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0answers
98 views

Derive mean and variance from equation

i have given a simplyfied one-dimensional Fokker-Planck equation : $\psi(p,t)=\frac{1}{\sqrt{2\pi vt}}\exp(-\frac{p^2}{2vt})$ My thoughts : ok, this looks pretty similar to the gaussian distribution ...
2
votes
2answers
200 views

Given $X$ is distributed normally, is $Y=aX$ normal? (for some constant a)

I assumed it was true, and then found $f_Y(y) = f_x(\frac{y}{a}).a^{-1} = a^{-1}\frac{1}{\sqrt{2\pi\sigma^2}}.\exp\{-\frac{(\frac{y}{a}-\mu)}{2\sigma^2}\} = ...
2
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1answer
162 views

Sample Mean & Variance

Let $X, Y$ be IID $\sim N(\mu, \sigma^2)$. $$M = \frac12(X + Y),\qquad V = (X - M)^2 + (Y - M)^2$$ Consider the joint moment generating function of $(M, X - M, Y - M)$, show that $M$ and $V$ are ...
1
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3answers
915 views

Generate a set of random numbers with an average evenly distributed between two given values

1) I generate 1000 random numbers between 0 and 10 and take the average. If I do the above action "many" times the resulting average values will be a normal distribution over 0 to 10. Correct? What ...
1
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3answers
142 views

Find standard deviation

The assignment is given: In a heat regulated room, we have two temperature limits $T_{\text{min}} = 16$ and $T_{\text{max}} = 24$. If the temperature is between $T_{\text{min}}$ and ...
4
votes
3answers
594 views

The probability density function of the ratio of two normal R.V.s

I'm looking for some help with this probability problem. Here's the question: Suppose that $X$ and $Y$ are independent standard normal random variables. Show that the probability density function ...
1
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2answers
7k views

Calculating mu and sigma (μ and σ) of a normal random variable

Let X be a normally distributed variable with unknown parameters μ and σ (sigma). If we know that P (X ≥ 75) = 0.7291 and P (X ≥ 83) = 0.7764. With the information given Is it possible to determine ...
1
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2answers
932 views

Approximating a sum of exponential distribution with a normal distribution

Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the ...
2
votes
1answer
60 views

Probability of being from a certain distribution

I was preparing for an exam in Image Analysis, when I found this probability problem in one of the old exam sets. A bunch of products is measured, and two features $(x_1,x_2)$ are measured on each. A ...