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
35 views

Normal distribution in equality

Let $p(x)=a_1x_1+a_2x_2+. . . .a_n x_n$ be a polynomial such that $\sum_ia_i^2=1,$ each $x_i \sim N(0,1)$. then we know that $p(x) \sim N(0,1)$. How can we bound $\Pr_{x\in ...
0
votes
2answers
70 views

probability normal distribution

A model for the movement of a stock supposes that if the present price of the stock is s, then after one time period it will be either (1.012)s with probability 0.52, or (0.99)s with probability ...
1
vote
1answer
61 views

Probability - normal distributions

The time it takes for a calculus student to answer all the questions on certain exam is an exponential random variable with mean 90 minutes. If all 100 students of a calculus class are taking that ...
3
votes
2answers
184 views

Covariance of a Normal with its Square

Assume there is a random variable distributed normal $X\sim N(\mu,\sigma^2)$. Is there an analytic expression for the covariance of $X$ with its square $X^2$? $$\operatorname{Cov}(X,X^2)$$ I have ...
0
votes
1answer
124 views

How to prove that $Y=\ln(X)$ approximately Normal when $X$ is a Normal random variable with $\mu\gg\sigma$

I wanted to prove that PDF of $Y=\ln(X)$ tends to a Normal distribution with $\mathcal{N}(\ln(\mu_{x}),\sigma^{2}_{y})$ when $X\sim\mathcal{N}(\mu_{x},\sigma^{2}_{x})$. It is also important to note ...
1
vote
0answers
111 views

Problem involving the bivariate normal distribution

If $X$ and $Y$ have a bivariate normal distribution with $\mu(x)=\mu(y)=0$, $\rho=0$, $\sigma(x)=\sigma(y)=10$. Find the following: A) The probability of getting a point $(x,y)$ inside the ...
1
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0answers
200 views

Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
0
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0answers
62 views

Using an appropriate hypothesis to test whether two means are different

Manager examined potential differences between two models of bicycles. The mean life of the bicycles is of primary concern. The followings table provides the available date which measured in ...
8
votes
1answer
289 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
2
votes
1answer
88 views

what is the joint distribution of these two random variables?

what is the joint distribution of two random variables,$∑_iX_iY_i$ and $(∑_iX_i)(∑_jY_j)$? Note that since $n$ is a large number and all the random variables are $iid$, using central limit theorem, ...
-1
votes
3answers
291 views

Variance of transformed random variable

The relationship of two random variables is given by $$ X = \Phi(Y) \Leftrightarrow Y = \Phi^{-1}(X),$$ where $\Phi(\bullet)$ is the standard normal cdf and $\Phi^{-1}(\bullet)$ the inverse of the ...
1
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0answers
218 views

Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
1
vote
1answer
95 views

Does $0$ correlation imply independence for marginally normal distributions?

Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$. Can someone give a hint why this is true ?
1
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3answers
125 views

Let $ X_1,X_2,…,X_n$ be i.i.d. $N(\theta_1, \theta_2)$, please prove that $E[(X_1-\theta_1)^4] = 3\theta_2^2$

If $X_{1}$, $X_{2}$, ..., $X_{n}$ is sampled from $N(\theta_1, \theta_2)$, how can I prove that $E [(X_{1} - \theta_1)^{4}] = 3 \theta_2^{2}$? I started off this question finding the completely ...
-2
votes
2answers
235 views

Is first order moving average a Markov process?

Given first order moving average $$ x(n) = e(n) + ce(n-1) $$ where $e(n)$ is a sequence of Gaussian random variables with zero mean and unit variance which are independent of each other, and $c$ is ...
1
vote
1answer
43 views

Cumulative Distribution Function of $X = Y*I + Z*(1-I)$

I have a random variable $X = Y*I + Z*(1-I)$ where $Y~N(u, sigma^2)\,,\,\, Z~N(u, sigmac^2)$, and $I~B(1,1-p)$. What I can't seem to figure out is how to get a cumulative distribution function for ...
0
votes
1answer
40 views

Interval Estimation when $\overline{Y}$ and $S$ is unknown.

Question: A random sample of size $n=9$ is drawn from a normal distribution with $\mu=27.6$. Within what interval $(-a,+a)$ can we expect to find $\frac{\overline{Y}-27.6}{S/\sqrt{9}}$ $80$% of the ...
2
votes
1answer
200 views

Chance of two Gaussian distributions being observations of the same phenomenom?

For an algorithm used for generation of a road map based upon position-samples, I am looking for a method of determining the probability of a sample belonging to an already discovered element of the ...
1
vote
1answer
55 views

Independence of Combination of Normal Random Variables

$\newcommand{\Cov}{\operatorname{Cov}}$ I have a practice question I'm trying to answer in studying for an upcoming exam: $X\sim N(0,1)$ and $Y\sim N(0,1)$ and I have $\rho(X,Y)=0.4$. Define a ...
1
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0answers
35 views

algorithm to use to balance a set of IPs into a set of buckets

So we have a set of IP addresses (~3000) and want to balance them into 4 different buckets. What we are doing now is very simple by treating the last part of the IP as integer and mod it by 4. e.g. ...
0
votes
1answer
70 views

What does $E[{\bf{x}} {\bf{x}}^{T}]$ mean?

It's known that $E[{\bf{x}} {\bf{x}}^{T}]={\bf{\mu \mu}}^{T}+{\bf{\Sigma}}$ but I have seen a very similar identity using data points $\bf{x_{n}}$ and $\bf{x_{m}}$ sampled from a multivariate Gaussian ...
1
vote
1answer
225 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
2
votes
1answer
418 views

What is the analytic expression for PDF of joint distribution of two Gaussian random vectors?

I know that if $X$ and $Y$ are random variables with respective PDFs, $$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left\{-\frac{\left(x-\mu_x\right)^2}{2\sigma_x^2}\right\} \\ f_Y(y) = ...
2
votes
1answer
340 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
1
vote
1answer
943 views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
2
votes
1answer
101 views

Log-likelihood for multinominal normal distribution

Given $n$ jointly-normal random variables $X_1, X_2, \dots, X_n$, with $$\mu_i=\mu\forall i \in\mathbb{N}^+$$ $$\sigma_i=\sigma\forall i$$ $$\rho_{i,j}=\rho\forall i,j \mbox{ with } i\neq j$$ what is ...
0
votes
1answer
117 views

Histogram with Gaussian bell curve

How do I create/calculate the probability density curve in a histogram which is scaled to the frequency axis with ABSOLUTE values (example)? The curve should be based on the calculated average and the ...
0
votes
1answer
89 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
vote
2answers
513 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
186 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
99 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
286 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
184 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
26 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
827 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
228 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
620 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
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0answers
146 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
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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
168 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
97 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
235 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
91 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
173 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
849 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
4k 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 ...