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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16 views

How prove the distributions of statistic below [closed]

Prove of distribution function of $V=\frac{X_1-\bar{X}}{S}$ where $S^2=\sum_{i=1}^n \left(X_i-\bar{X}\right)^2$
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21 views

Calculating expected value and variance given joint Gaussian distribution

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X_i: \Omega \rightarrow\mathbb{R}$ random variables for $i=1,2$. $X_1$ and $X_2$ are jointly Gaussian distributed with joint ...
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11 views

Linear Regression Model Variation

I know that Linear regression allows us to model how an outcome variable Y depends on one or more predictor (sometimes called independent variables) $X_{1},X_{2},..,X_{p}$. Equivalently, the linear ...
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1answer
20 views

How to “denormalize” a standard deviation?

Lets say I have a vector of variables which have all been standard normalized with $(\mu_n, \sigma_n)$ by doing the operation $\frac{\mathbf{y} - \mu_n}{\sigma_n}$. I then have a model which predicts ...
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1answer
34 views

Covariance of $X$ and $Y^2$ where $(X,Y)$ is bivariate normal

I'm trying to solve a case where there is bivariate random vector $(X,Y)$ that has the bivariate normal distribution below ($-1<\rho<1$): $$\begin{pmatrix} X\\ Y \end{pmatrix}\sim N_{2}\...
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Restrict domain of variance in Gaussian distribution

Let's say I have a point $a \in \mathcal{U}$, where $\mathcal{U}$ is a set, and $x$ is a Gaussian random variable with mean $0$ and variance $\sigma^2$. Is it mathematically correct to say: Restrict $...
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1answer
29 views

product of normally distributed random variable with discrete random variable [closed]

I have to solve the following problem, which doesn't look difficult to me but I can't seem to solve it: Let $N$ be a real valued Gaussian random variable with mean zero and variance $1$ and let $Z$ ...
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42 views

Problem with module

Knowing that… $Y:=\left\{\begin{matrix} X & if & |X|<a\\ -X & if & |X| \geq a \end{matrix}\right.$; $X \sim N(0,1)$; $a>0$; …and obviously $|X|:=\left\{\begin{matrix} X & ...
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3answers
31 views

Distribution of $\frac{X_1X_3+X_2X_4}{X_3^2+X_4^2}$

$X_1, X_2, X_3$ and $X_4$ are independent standard normal random variables. Find the distribution of $$T=\frac{X_1X_3+X_2X_4}{X_3^2+X_4^2}$$ I have found that $U=X_1X_3+X_2X_4$ follows standard ...
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1answer
29 views

A Normal distribution inequality

Denote by $\phi(z)$ and $\Phi(z)$ the density and distribution of a standard Normal variable (mean=0, sd=1), respectively. It appears that for $z > 0$, $$ \frac{\phi(z)}{1 - \Phi(z)} < z + \...
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1answer
39 views

Confidence interval $N(\theta,\theta)$

Let $X_i$ be i.i.d. r.v. with $N(\theta,\theta)$ I calculated $$E[\bar{X_n}] = \theta$$ $$Var[\bar{X_n}] = \theta/n$$ And want to construct a confidence interval $I_{\theta}$ that is centered around ...
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1answer
22 views

Expectation of Inverse Normal CDF

Suppose a r.v. $\mu$ is distributed Normal $N(\theta,\sigma^2)$. Is there any way to derive the expectation $\mathbb{E}(\frac{\mu}{\Phi(\mu)})$ where $\Phi$ is the CDF of a standard Normal random ...
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1answer
21 views

$f, g$ are probability density functions of an normal distribution N(0,1), prove h is $N(0,\sqrt 2)$

I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function. When $f$ and $g$ are $exp(\lambda)$ it's solved by $\int_{0}^{...
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20 views

If $t\mapsto X_t$ is continuous almost everywhere and $(X_t)$ has independent increments, then $X_t - X_s$ follows a normal distribution?

The following statements can be found at Glasserman's Monte Carlo Methods in Financial Engineering. Given a stochastic process $(X_t)_{t\in [0,T]},$ if the mapping $t\mapsto X_t$ is continuous ...
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1answer
33 views

Asymptotic rate of decrease of error function

The complementary error function is defined as $$ \text{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}}\int_0^{x} e^{-t^2} dt $$ and is related to the Gaussian (Normal) distribution. Is there an approximation of ...
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14 views

Any normal random variable is ''essentially'' surjective

Let $\Phi$ be the cumulative distribution function of the standrad normal distribution. Denote $X: (0,1) \rightarrow \mathbb R$ its inverse. Then $X$ is a standard normally distributed random variable ...
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1answer
24 views

Should I be using normpdf to answer this question?

If the mean of a sample is 30, and the standard deviation is 10, then how would I evaluate a question that asks me how likely it is that I would obtain a value of $34$? Also the size of the sample is ...
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27 views

A question on hypercubes and the central limit theorem

I was reading a book on Monte Carlo methods and now I'm trying to make sense of an excercise. At one point they say that according to the central limit theorem most of the points ${\bf x} \in [0,1]^d$ ...
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1answer
37 views

PDF of $Q$ Random Variable

Let $X\sim N(0,25)$, $Y\sim N(10,100)$, $Z\sim N(-10,50)$ and $Q=\tan^{-1}\left(\frac{Z}{\sqrt {X^2+Y^2}}\right)$ When I simulate $Q$ random variable with Monte Carlo method, I'm getting this ...
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28 views

Covariance of two cdfs of two standard normal distribution [duplicate]

Let X1 and X2 be standard bi-gaussian distribution with correlation ρ: X1~N(0,1), X2~N(0,1), cov(X1,X2) = ρ Then Y1 and Y2 are cdfs of X1 and X2, we know they are random variables of uniform ...
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2answers
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Are there independent random variables $η_1 = ξ_1 + ξ_2, η_2 = ξ_1 - ξ_2$ if $ξ_1,ξ_2$ are independent?

The random values $ξ_i, i = 1, 2$, are independent and have a standard normal distribution. Are there independent random variables $η_1 = ξ_1 + ξ_2, η_2 = ξ_1 - ξ_2$? I'm tried to find the ...
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2answers
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A statistics problem (normal distribution) [duplicate]

If $X\sim N(0, \sigma^2)$, how can I compute $\operatorname{Var}(X^2)$? Here is my idea... but I cannot get there. $$\operatorname{Var}(X^2) = E(X^4) - (E(X^2))^2$$
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Maximum of a sequence of chi-square RV

Let $Z_{1}, \ldots, Z_{n}$ be i.i.d. $N(0,1)$ random variables, and define $X_{j}=Z_{j}^{2} .$ Show that$\mathbb{E}\left[\max _{1 \leq j \leq n} X_{j}\right] \leq c \log (2 n)$ for some absolute ...
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94 views
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Change of variable for conditional probability

Suppose I have random variables $X$, $Y$ and $Z$, with $Z \sim N(0, \sigma^2)$ and $Y = kX + Z$, I am looking for a proof of the fact that $f_{Y\mid X}(y\mid X = x) = \frac{1}{\sqrt{2 \pi} \sigma} \...
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2 views

Alternative way of modelling item response theory

I will assume one already knows about IRT. Consider that 2-parameter irt model $$ P(Y_{ij}=1|\theta_j,a_i,b_i)=p_{ij}={\frac {1}{1+e^{-a_{i}({\theta_j} -b_{i})}}}$$ in Bayesian analysis we set $\...
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27 views

How do I determine the distribution?

The question is as follows: Let $X_1, . . . , X_{2n}$ be a random sample of $2n$ observations from a standard normal distribution. Let $Y=\frac1n \sum_{i=1}^n X_i$ denote the average of the first $n$ ...
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25 views

Conditional distribution of $X_{1}+3X_{2}-2X_{3}$ given that $2X_{1}-X_{2}=1$ $X_{1},X_{2},X_{3}$ i.i.d $N(2,1)$

From the book An Intermediate Course in Probability by Allan Gut, second edition. Let $X_{1},X_{2},X_{3}$ be independent, $N(2,1)$-distributed random variables. Determine the distribution of $X_{1}+...
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25 views

Bivariate normal distribution aX+bY [closed]

$(X, Y) \sim$ Bivariate Normal I want to find out probability function of $aX+bY$ without using the moment generating function.
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1answer
51 views

What's the distribution of $xy+xz+yz$ where $x,y,z $ are independent standard normal?

We know the product of two independent Normal random variables has a normal product distribution, or Variance Gamma distribution if they are correlated. But, what if there are three Normal random ...
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30 views

$X,Y\sim \mathcal{N}(\mu, \sigma^2) \implies X+Y$ and $X-Y$ are independent.

Use the following result: Let $X,Y\sim \mathcal{N}(0,1)$ be independent random variables and let $v,w\in \mathbb{R}^2$ two orthogonal vectors s.t. $||v||=||w||=1$. Then $V =v·(X, Y ), W = w·(X, Y)$ ...
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2answers
21 views

how is the sum of 2 normally distributed random variables different to 2 times a normally distributed variable

Say we have X~N(10, 100). It seems to hold that X+X~N(20, 200), however, if we multiply X with constant we have to multiply the variance with the square of the constant. Take for example 2, then we ...
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1answer
28 views

Distribution of the sum of Brownian motions

I believe that for a standard Brownian motion $W(t)$, $W(t)+W(s)$ has a normal distribution with mean $0$ and variance $s+t$ (because they are two independent normally distributed variables)? But is ...
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11 views

How to rearrange data to fit specific normal distribition ~ (u, sig2) , without changing total amount?

Background I have a kind of product need to distribute to 42 stores . The total number is different in everyday depend on the supplier . I have a distribution algorithm which make sure (stock_number +...
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1answer
19 views

Find $c$ such that $P(Z^2 > c) = 0.95$ [closed]

I was wondering if any of you could help me with this statistics problem.
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28 views

Dice rolled 40000 times, let X be number of times 3 is rolled, find P(10000<X<12000)

I've been going through past exam papers and I'm not sure how to do this. Let X be the number of times the outcome 3 comes up in 40000 throws of a fair astragalus. (You can assume that an astragalus ...
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1answer
12 views

Variance of a linear transformed standard normal r.v.

I have the sequence $X_i$ that converges to $N(0, 1)$ in distribution and $$Y = 2X_i + 1$$ I was able to find $E(Y) = 1$, but I am struggling at $Var(Y)$. From the general variance formula $E[(X-\mu)...
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18 views

Calculating probability that a certain % of samples fall between a range

The height of students is distributed with an average of $186.7\; cm$ and an SD of $5.6\;cm$. 200 samples of 50 students (i.e. 200 groups of 50 students) were taken from the population, and the ...
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1answer
23 views

What mean does the machine have to be set on to pass the test?

say we have a machine producing packs of butter, the weight of the packs are normally distributed with X~N(u ; 9). The packs go through a test where a sample of 25 is picked, and if the sample has a ...
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1answer
68 views

What is a formula to widen/restrict a distribution of values?

This is a beautiful Monday morning for problem-solving! Here is what I am trying to achieve. I am studying an electoral model and I want to analyse its behaviour using different settings. One of ...
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1answer
14 views

How to find the mean of this standard distribution?

Question: The probability distributions for 2 variables are defined as follows$ X$ ~ $N$$(120, σ^2)$ and $Y$ ~ $N$$(μ, 2σ^2)$ and $P(X < 124)$ = $P(Y > 124$). Calculate $μ$. I tried this for ...
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6answers
106 views

Is there a real valued positive function such that it and its square integrate to $1$

Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f > 0$ and $$ \int_{-\infty}^\infty f(x) dx = \int_{-\infty}^\infty f(x)^2 dx = 1. $$ I suspect the answer is yes. I ...
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3answers
58 views

Distribution of $\Big(Y_1+Y_2\Big)^2$ and $\Big(Y_1-Y_2\Big)^2$ where $Y_i \sim N(0,1)$

Does anyone know what is the distribution of $(Y_1+Y_2)^2$ and $(Y_1-Y_2)^2$ where $Y_i \sim N(0,1)$ are independent variables? I have tried to go through the joint pdf, but when trying to change ...
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1answer
35 views

Proof of linear transformation of random variable

Picture This is a part of proof of linear transformation of normal random variable, can anyone tell me how to get procedure 4 from procedure 3?
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1answer
17 views

Variance of Linear Combination of Standard Normals

I have a random variable $X_n=\frac{1}{n} \sum^n_{i=1} Z_i$ for $n\in \mathbb{Z}^+$ where $X_0=0$ and the $\{ Z_i \} \sim N(0, 1)$ for all $i$. The $Z_i$ are independent. I need to find the ...
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1answer
32 views

Quick Question: Variance of a two related Gaussian distributions

I have two random variables, $X$ and $Y$. Both follow a Gaussian distribution, and $$X \sim N(0,1)\;.$$ After some manipulation, I got that $$P(X \leq z) = P(kY \leq z)$$ where $k$ is some constant. ...
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1answer
43 views

How does $\arccos$ appear in the expectation of a normal random variable?

I found the following equation in a paper: $\mathbb{E}_{w \sim N(\mathbf{0},\mathbf{I})} [\mathbb{I}\{w^\intercal x_i \geq 0, w^\intercal x_j \geq 0\}] = \dfrac{\pi - \arccos(x_i^\intercal x_j)}{2\pi}...
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2answers
24 views

“Inverse” moment generating function of standard normal distributed random variable

This is just a trivial question maybe but, is the Moment generating function for $X$ the same as for $-X$ for a normally distributed random variable, so $E(e^{tX})=E(e^{-tX})$? If not, what is the ...
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1answer
25 views

Anti-concentration for Gaussian

Is there a reasonable anti-concentration bound for Gaussian? Let $X\sim\mathcal N(0, \sigma^2)$, can we get $P(|X|>\epsilon)>1-\delta$? Thanks.
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0answers
14 views

Can someone explain why we have this for a GP regression conditioned on the observations

Consider a Gaussian Process (GP) regression $y_t=f(x_t)+\epsilon_t$ with iid noise $\epsilon_t \sim N(0,\sigma^2)$. Can someone please explain why conditioned on $(y_1, \ldots, y_{t-1})$, $\{x_1, \...
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1answer
25 views

Find the MoM estimator of $\theta$ when $X_i \sim N(a_i\theta, 1)$

Let $X_1, ..., X_n$ be independent random variables on some probability space such that for each $i = 1, ..., n$ we have that $X_i \sim N(a_i\theta, 1)$, where $a_1,...,a_n$ are given constants ....

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