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

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0
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1answer
105 views

What is the distribution of an unconditioned random variable knowing the conditional distribution?

I have two random variables $X$ and $Y$. I know that $Y$ can be approximated by a $N(\mu_1,\sigma_1^2)$ distribution (in particular $Y$ is not negative) and I also know that $X|Y \sim N(a+bY,c+dY)$ ...
1
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2answers
580 views

The correlation between two normal distribution

Let $X$ have the $N(0,1)$ distribution and let $a>0$, show that the random variable $Y$ given by $$Y=\begin{cases} X & \text{if }|X|<a\\[5pt] -X &\text{if }|X|\geq a\; \end{cases}$$ has ...
2
votes
1answer
4k views

Prove Variance of a normal distribution is (sigma)^2 (using its moment generating function)

Prove that the Variance of a normal distribution is (sigma)^2 (using its moment generating function). What I did so far: Var(X) = E(X^2) - (E(X))^2 E(X^2) = Mx'(0) = r/(sqrt(2pi)*sigma) * ...
4
votes
1answer
194 views

Expectation value of $1/x$

Given a random variable $x$ which is assumed to follow a Gaussian distribution $x \sim N( \mu, \sigma^2 )$ and $x$ is further known to be positive, I am interested in the following expectation value: ...
1
vote
1answer
101 views

Mentally Estimating the Normal CDF

More than once I have seen this sort of frustrating question on a Mathematics GRE practice test: A fair die is tossed 360 times. The probability that a six comes up on 70 or more tosses is... a) ...
0
votes
1answer
1k views

Finding Mean Value and Standard Deviation

The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than ...
4
votes
2answers
328 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
0
votes
1answer
197 views

Distribution of Product of Random Variables with one being the normal distribution.

Let X and Z be independent, with $X\sim N(0,1)$, and with $\textbf{P}(Z=1)=\textbf{P}(Z=-1)=\frac{1}{2}$. Let $Y=XZ$ (i.e., Y is the product of X and Z). (a) Prove that $Y\sim N(0,1)$. (b) Prove ...
0
votes
0answers
27 views

Normal distribution interval

I am trying to find $c$ so that $P(|X - 5| < c) = .95$ with $\mu = 5, \sigma^2 = 4$. I came up with: $P( \frac{-c}{2} < Z < \frac{c}{2}) = .95$ Attempting to solve for $c$, the $z$-score ...
2
votes
0answers
973 views

Standardized Normal Distribution Problem

Mopeds (small motorcycles with an engine capacity below $50~cm^3$) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the ...
0
votes
1answer
256 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
1
vote
1answer
780 views

Determining The Value, c, A Random Variable Assumes

The question I am working on is: In each case, determine the value of the constant c that makes the probability statement correct. $P(c \le |Z|)=0.016$ Here is my attempt: $P(|Z| \ge ...
3
votes
0answers
147 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
0
votes
0answers
34 views

Covariance calculation for mixture Gauss

This equation what I found on the wikipedia is a bit strange for me. How I can compute in matlab? If example the size(x) = [1000, 2](2 dimension gauss) then $size(\mu) = [1, 2]$ for each cluster. ...
1
vote
1answer
176 views

statistics: probability, normal distribution

The time that customers take to complete their transaction at a money machine is a random variable with mean $\mu$ = $2$ minutes and standard deviation $\sigma$ = $0.6$ minutes. About 30% of ...
-1
votes
1answer
106 views

Find the warranty period such that the battery is replaced under warranty 0.5% of the time

Problem The mean life of a Chevy Volt battery (normally distributed) is $1000$ hours and the standard deviation is $100$. How many hours should GM warranty the battery for so that it has to replace ...
0
votes
1answer
65 views

Can the the multivariate normal distribution be one dimensional?

Can the the multivariate normal distribution be one dimensional? Or should you then just use the normal distribution? I mean does an one dimensional multivariate normal even make sense?
0
votes
1answer
254 views

Calculation of mean value of normal distribution if we only know the maximum and minumum value

If I have only maximum and minimum values, can I calculate mean value of normal distribution?
1
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0answers
101 views

CAPM-model - necessary conditions for BETA to be a parameter in the conditional expectation

CAPM-model - necessary conditions for BETA to be a parameter in the conditional expectation between the real return on the asset and the stock market return. Okay, trying to be more explicit: Let ...
2
votes
1answer
201 views

Expected value of $xx^{T}$ for multidimensional Gaussian

I need a bit of help understanding a step in the derivation of the expected value of $\bf{x x^{T}}$, that is, $E[\bf{x x^{T}}]$ with a Gaussian distribution. By definition, using the D-dimensional ...
3
votes
3answers
1k views

7.7 standard deviations away from the mean?

I'm confused. I have a problem where I have to find the probability that x is below the z value 7.7. My z table only goes to z values of 3.4. How do I calculate this? These are the hints my teacher ...
2
votes
3answers
5k views

Standard deviation of the weighted mean [duplicate]

How do you find the standard deviation of the weighted mean? The weighted mean is defined: $\bar{x}_w = \frac{\sum{wx}}{\sum{w}}$ The weighted standard deviation (since it is not specified, I take ...
1
vote
1answer
9k views

Find a Probability of a Normally Distributed Random Sample

Please help me figure out how to do this problem. I need to be able to understand how to solve problems like this. Thanks times a million! Problem: An employer is interested in the commute times for ...
1
vote
1answer
490 views

Rayleigh distribution

I have this question from my statistical theory course: A sniper shoots at a target. X and Y measure its deviation on the x and y axes. X and Y are independent and are distibuted normally with mean=0 ...
1
vote
1answer
194 views

What is the physical meaning of the output/ y -value of a normal distribution? (not the area under its curve)

Forgive me for my lack of knowledge regarding math terminology. I'm learning basic statistics right now, and I can see pretty intuitively that the area under a normal distribution on a certain ...
0
votes
2answers
45 views

Algorithm for integral of standard distribution

I need help in producing random data that follows standard distribution. Since it is to be used in a computer application, I would prefer an algorithm before a table. So, this is what I need. The ...
0
votes
1answer
115 views

Weibull Distribution question

Not sure how to approach this one, some help would be appreciated. It has been observed that 5% of the students who take a certain exam will finish in less than 20 minutes and 95% will finish in ...
1
vote
1answer
121 views

what is the covariance matrix for deterministic signal+normal noise

Say that we have a signal that is written as follow $y=y_0+r$ where $y$ and $y_0$ are n-dimensional vectors and $r$ is n-dimensional noise vector. I would like to have $r\sim \mathcal{N}(0,\Sigma)$ ...
0
votes
1answer
77 views

Gaussian random variable(GRV)

$X$ is a Gaussian random variable $N(2,2)$. Also given are values x1=1 and $x=3. i. Write a program to calculate the probability Pr(x1 ≤ X ≤ x2). ii. Write a program to calculate the probability Pr(|X ...
0
votes
0answers
39 views

How to calculate t-values from percent ranks?

A table from a psychological test is giving percent ranks (PR). How can I transform those into t-values by calculation? (Assuming normal distribution.) I found a lot of references on how to ...
0
votes
0answers
96 views

How to put a bivariate normal distribution under standard form

If I have the following gaussian integral: $$ \int_{-\infty}^{a}\int_{-\infty}^{b}\exp\left[-\alpha x^2+\beta x-\theta y^2+\gamma y+2\lambda xy\right] dxdy$$ ,where X and Y are standard normal R.V. ...
1
vote
1answer
80 views

About the differential entropies of well-known continuous distributions

Assume that the continuous random variable $X$ has a distribution (in a closed form expression) with differential entropy $h(X)$. Q) Then, is it true for any continuous distribution that the ...
2
votes
0answers
109 views

Can any one help me normalize this equation? (Modified 3D Gaussian)

$$\exp\left( - e^{d-sz} - 2 \left( \frac{z^2}{r^2f^2}+\frac{x^2+y^2}{r^2} \right) \right)$$ Note if this equation can't be normalized another equation with similar proprieties would also be ...
0
votes
1answer
300 views

Maxwell-Boltzmann velocity PDF to CDF

I need to draw from a Maxwell-Boltzmann velocity distribution to initialise a molecular dynamics simulation. I have the PDF but I'm having difficulty finding the correct CDF so that I can make random ...
1
vote
2answers
107 views

incrementally calculate gaussian distribution

Is there a way to incrementally calculate the parameters for a Gaussian distribution? The only parameters that are of interest are the mean and standard deviation, not the height (ie. normal ...
0
votes
1answer
103 views

Efficient method of approximating a distribution with Gaussian

Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$), how to find the best ...
2
votes
1answer
88 views

Dirac function and integration by parts

I have some problems to show the following relation, apparently using integration by parts and knowing that $\phi$ denotes the density of the standard one dimensional normal distribution. $$\int ...
2
votes
1answer
179 views

What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
0
votes
4answers
848 views

Why this “interpolation” is correct?

Why does the interpolation in this image make any sense? Why won't I take $N(6.3) - N(6.1)$ or $N(6.2) - N(6.1)$? It is not linear at all.
0
votes
1answer
63 views

Parameter estimation with GMM

I have estimated the parameters of normal distribution with GMM and got the following results: $mean = -0.01168 , p-value = 0.83519, Sd = 1.77 , p-value = 0.00000.$ I'm bit confused in ...
1
vote
1answer
2k views

Variance for a product-normal distribution

I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution. It's a strange distribution involving a delta ...
0
votes
1answer
208 views

Does an independent-increment Gaussian process necessarily have Gaussian increments?

Suppose a stochastic process is both independent-increment and Gaussian. Are all its increments Gaussian distributed? Thanks!
0
votes
1answer
128 views

Dividing data of a given distribution

Having a dataset with distribution $p$ (e.g. uniform or normal), if we divide the dataset into $n$ parts with equal size, is it valid to say that each part still has distribution $p$?
2
votes
2answers
392 views

What is the probability that two samples represent the same normal distribution?

Yes, it's a basic question. But, I have searched about 25 web pages for this and found only things that were irrelevant or incomprehensible. So I have indeed tried. My question is: I have two ...
-2
votes
1answer
80 views

For a normally distributed random variable, find a value from given tail probability

Problem Let $ X \sim N(65,20) $. Find correct to $3$ Decimal Place the value of $x$ such that $Pr(X>x) = 0.43$. Progress I've gotten to $\frac {x-65}{2(5)^{1/2}} =0.1764$ and hence $x = 67.789$? ...
0
votes
1answer
216 views

Percentage point of Normal Distribution.

Let $$X \sim N(65,64) $$ Find the lower $2$% point for $X$; that is, find the value of $x$ such that $Pr(X<x) = 0.02 $ i know i need to do something like $\frac{X- 65 }8 = ... $ but not ...
0
votes
0answers
199 views

Statistics Question - Normal Distribution

The scores of a final exam have a Normal Distribution with mean $75$ and standard deviation 6. An independent sample size of $9$ is drawn from this distribution. The corresponding random variables are ...
3
votes
0answers
67 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
votes
1answer
105 views

Inverse of a sub-matrix

I have a multivariate Gaussian distribution with known $\mu$ and $\Sigma$. I want to evaluate it given a vector $x$. However, some of the attributes of this vector may be unknown, in which case I want ...
0
votes
1answer
113 views

How can I solve this integral?

How can I solve the following integral? $$\int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} ...