# Questions tagged [normal-distribution]

This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

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### Find relationship between normal and stanard normal

I have no idea about how should I proceed this question . QAQ
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### Bound on error when Poisson distribution is approximated by normal distribution

At several resources I found the claim that for large $n$ the Poisson distribution with mean $n$ is approximately normal with mean and variance $n$. Quite a few of these resources quote the central ...
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### Sum of normal distribution combined with a Bernoulli process

Imagine this scenario: 50% of the time I sample from a Normal distribution with $\mu$ and $\sigma$ while the other 50% I sample from a distribution with $-\mu$ and $\sigma$. However, I know that 50% ...
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### Statistics question asking for the standard deviation of a data that has a normal distribution [closed]

Basically the question states: MENSA is an organization whose members have IQs in the top 1% of the population. IQs are normally distributed with mean 100, and the minimum score required for admission ...
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### Diffrence in probability distributions of sepertaed groups

If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and ...
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### In the 1992 presidential election Alaska's 40 election districts averaged 1886 votes per district for President Clinton The standard deviation was 600 [closed]

In the 1992 presidential election, Alaska's 40 election districts averaged 1886 votes per district for President Clinton. The standard deviation was 600. (There are only 40 election districts in ...
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### Proof of Independence of Sample mean and sample variance

This concerns Section 7.8.2 in the book A First Course in Probability by Sheldon Ross 10th Edition. Section 7.8.2 says that : Let $X_1, X_2.\ldots X_n$, be independent normal random variables each ...
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### Does "independent variables X and dependent variable Y are jointly gaussian" means "the residual term has 0 conditional mean"?

I saw the following statement in my lecture note: "The data generation process is $y = x+\epsilon$, whereas in the regression we run y on x so the regression model is $y = \beta_{OLS} x+e$, the ...
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### Compute the probability of multivariate Gaussian distribution

I am reading a proof from a paper: Zhu, Sicheng, Xiao Zhang, and David Evans. "Learning adversarially robust representations via worst-case mutual information maximization." International ...
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### Sampling Normal Distribution; Box-Muller, Inverse Transform, Rejection, Approximations?

I assume $X\sim\mathcal{N}(\mu,\sigma)$ and wish to sample values but I am confused about different approaches and concepts that seem to be relevant for this problem. It appears to me that this ...
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### Exponential distribution from Standard normal distribution

How do we get part (c)? I tried looking for relationships between the chi square distribution and the exponential distribution but couldn't find anything.
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### Conditional density calculation

If we know the density of a vector $x$ is $f(x)$, which could be multivariate standard normal $N(0,I_r)$, and we know the conditional density of $f(y|x)$, which is multivariate normal $N(Ax,I_T)$ with ...
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1 vote
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### A bit of confusion with the Gaussian measure and pushforward by a random vector

Let $d\mu$ be a centered Gaussian measure on $\mathbb{R}^n$ with the covariance matrix $\sigma$ and $X : \mathbb{R}^n \to \mathbb{R}^n$ be any measurable mapping. Or we can simply regard $X$ as a ...
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1 vote
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I want to calculate the inverse Fourier transform of $\exp\left[-e^{-k^2}\right]$. One of the ways I can imagine is to expand the exponential in a series $$\exp\left[-e^{-k^2}\right] = \sum_{n=0}^\... 0 votes 0 answers 35 views ### What is the minimum score to get a scholarship? The test scores of an entrance examination are normally distributed with a mean 230 and standard deviation 80, the cutoff for admission was 310 what % of examinees got admission, further, the ... 1 vote 0 answers 37 views ### Show that (M_1,\cdots,M_p) and (N_1,\cdots,N_q) are independent Let (\Omega,\Sigma,\mathbb{P}) be a probability space and X_1,\cdots,X_n:\Omega\to\mathbb{R} be independent normally distributed random variables. Show that if M_1,\cdots,M_p,N_1,\cdots,N_q\in\... 1 vote 0 answers 22 views ### Closed form expression for the minimal width interval that satisfies coverage on log-normal distribution I am interested in finding the minimum width interval (b_l, b_u) for b_l,b_u \in \mathbb{R} that obtains a coverage level of \alpha \in (0,1) for the log-normal distribution. The CDF of the log-... 0 votes 0 answers 22 views ### Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns Let us define the M \times N matrix \boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right], where \boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right) (i.... 3 votes 0 answers 65 views +50 ### Confusion with Complex Gaussian process with Auto-covariance I have a complex sequence z(t) in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ... 4 votes 2 answers 111 views ### fully customizable periodic function? I am looking for a bell-shaped periodic function f(x) with parameters a and b, with following characteristics: ( not sure if such function already exists or one can formulate one ) : oscillating ... 0 votes 0 answers 36 views ### How to prove x^T\Sigma x is quadratic? How to prove x^T\Sigma x is quadratic? where \Sigma is covariant varible(symmetric matrix) I did simple calculation with 3x2 matrix , 2x2 covariance, 2x3 matrix. the result is 3x3 matrix. But how ... -1 votes 0 answers 19 views ### Mean of sample generated by Exp and Norm I have the following scenario: i have a next event simulation in which I generate an arrival process in time slot [8:00 am, 11:00 am]. The inter-arrival time is assumed exponential with arrival rate 2 ... 3 votes 2 answers 135 views ### A tight upper bound for Gaussian integral. Consider two positive real number \mu and \sigma. Let m = 1, 2, \ldots be the natural number, I want to find a tight upper bound for the following part Gaussian integral:$$\int_0^{\infty} \exp \...
Let $v\sim N(0,\sigma_v^2)$, let $z, y_1, \dots, y_n\sim N(v, \sigma^2)$ be i.i.d stochastic variables. Calculate $E[v\mid z, z \geq y_i \ \forall i \in \{1,\dots , n\}]$ Said in another way, then ...
Lets define the following probability density distributions as: \begin{align} p(θ) &= N(θ; 0,1) \\ q(θ) &= N(θ; μ,σ^2)\\ p(y|θ,x) &= N(y; θx,σ_n^2) \end{align} where $N(x; m,v)$ ...