Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [normal-distribution]

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

0
votes
0answers
19 views

Conditional variance when there is uncertainity about the distribution of condition

Assume that $X \sim N(0,\sigma_x^2)$. Y has the following form \begin{align} Y &= \begin{cases} Y_1 \sim N(0, \sigma_1), & \text{w.p.} \quad \mu \\ Y_2 \sim N(0, \sigma_2), & \text{w.p.} ...
-1
votes
1answer
33 views

Integral of a standard gaussian distribution

Please see the image. I know that the integral of P(x) tends to 1. But the quadratic equation next to P(x) seems confusing.
1
vote
1answer
16 views

What is the probability distribution of this AR(1) function?

I'm preparing the exam for "stochastic models" and I encountered this exercise which is giving me a lot of problems: Let $X_t \sim AR(1)$, with $$X_t=-0.8X_{t-1}+ \epsilon_t, ~~~~~~~~~~\epsilon_t \...
2
votes
1answer
26 views

How to derive the formula for the expected value for maximum of n normal random variables

This is a follow-up question on this one: Expected value for maximum of n normal random variable @RobertIsrael states the following: Presumably the $X_i$ are independent. If $\Phi$ is the ...
2
votes
1answer
17 views

How should I calculate mean CI - from raw data or mean values?

I have experimenatal data from medical industry. In the experiment 4 observations were weighted, each one 2 times for more precise measurement. As a final weight, a mean from each pair is reported. ...
0
votes
0answers
7 views

The relation between $\bar y$ and $\Vert P_v Y \Vert$

In "Multivariate Analysis" I faced A lot of questions contain form of projection of a vector I should transfer it to be in terms of Y or $\bar y$ , see following example: Q: Let Y has a $N_n (\mu , \...
2
votes
1answer
27 views

For $X\sim\mathcal{N}(0,1)$, find measurable functions $g,h$ s.t. $g(X),h(X)$ are uncorrelated

Let $X\sim\mathcal{N}(0,1)$ a random variable. Find measurable function $g,h:\mathbb{R}\to\mathbb{R}$ such that $g(X),h(X)$ are uncorrelated. I would like to confirm my answer: Let $g,h:\mathbb{R}\...
-1
votes
1answer
20 views

Find a measurable function $g:\mathbb{R}\to\mathbb{R}$ s.t. $\mathbb{E}(g(\mathcal{N}(0,1)))=2$

Let $X\sim\mathcal{N}(0,1)$ a random variable. Find a measurable function $g:\mathbb{R}\to\mathbb{R}$ such that $\mathbb{E}(g(X))=2$.
0
votes
0answers
12 views

How to split a data set with two distributions

I have two normal distributions with equal sizes mixed together into one data set, it is assumed both have similar std, only the mean of both distributions is different. What will be easiest way to ...
0
votes
0answers
11 views

Estimate the probability distribution of two parameters

I'm currently facing a new type of problem, and i have no idea how to solve it, so any suggestion will be really appreciated ! The problem is the following: I have a matrix of temperatures, depending ...
0
votes
0answers
22 views

Show that sum of these two Random Variables is conditionally normal distributed (from IGARCH model)

According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed. $σ_t^2 = ...
-1
votes
0answers
8 views

How to normalize intra-day volume?

I'm new to this site. I'm trying to normalize call volume with a 30 minute interval for the day. What I do is get the per interval volume allocation by getting the value for that interval divided by ...
0
votes
2answers
17 views

Does the Integral over the product of distributions result in another distribution?

In this post one guy asks why those likelihoods in Maximum Likelihood get multiplied. It's clear why they get multiplied. I have another problem. In my textbook it is state that for each sample the ...
1
vote
1answer
32 views

Let $Z\sim N(0,1)$. Give a clean and rigorous proof that $Z^3$ cannot be normally distributed

Just as the title suggests. I simply have no idea how to prove that a random variable is not normally distributed. I'd really appreciate some hint.
0
votes
0answers
3 views

normalizing by using gaussian distribution for negative and positive numbers and feed in Min,Max normalization

I'm dealing in Python with a dataset which has 6 million float numbers belongs to 3 main parameters A, B , C and I map them in 24x20 matrices for each cycle and I plot them 480-values by 480-values. ...
1
vote
0answers
26 views

Derivative of sum of Gaussians wrt. sigma - How does this work?

Let $\varepsilon_i \sim \mathcal{N}(0,\sigma_i^2)$ be samples from independent Gaussians, we know by additivity of independent Gaussians that $\varepsilon_1 + \varepsilon_2 = \varepsilon'$ where $\...
1
vote
1answer
42 views

Linear combination of normal distribution which is not normal

Let $\xi_1, \xi_2$ be i.i.d N(0,1). Define $(X_1,X_2)=\begin{cases} (\xi_1, |\xi_2|) \quad \xi_1 \geq 0 \\ (\xi_1,-|\xi_2|) \quad \xi_1 < 0 \end{cases}$ This means we can rewrite $X_1=\xi_1$ and ...
0
votes
1answer
14 views

Conserving variance of normally disributed vector in linear transformation

I have normally distibuted vector $X$ with known variance $Var(X)$ and mean $\overline X$ and then I take new vector $Y$ as $W*X$ where $W$ is normally distributed matrix. If I required $Y$ to have ...
0
votes
0answers
27 views

Finding the probability of the sample standard deviation using Normal Distribution

The maximum wind speed on any given day in Dublin is modelled as Normally distributed with mean 10 knots and standard deviation 2 knots. (a) Explain potential deficiencies of using the Normal ...
0
votes
0answers
7 views

How to plot a data set on a normal distribution graph (what is the Y-axis)

I have a set of data based on wear of parts against number of hours run. From this I have calculated the mean, the standard deviation (population) and created the bell curve (normal distribution). Now ...
1
vote
0answers
32 views
+50

Markov chain converges to Normal Distribution. How to increase standard deviation

I am creating an MC (using the following recursive function). It is about a game that each side has different probs of scoring 1 point, 2 points, 3 points or no score (scoreDifference is the score ...
0
votes
0answers
14 views

What is unnormalised discrete distribution?

Can someone provide an easy to understand meaning of what unnormalized discrete distribution is and how is it different from normalized discrete distribution and when is unnormalized discrete ...
0
votes
0answers
15 views

Show that X is a sub-gaussian random vector with dependent sub-gaussian coordinates

Let $X \in R^n$ be a zero mean, random vector with sub-gaussian coordinates $X_i$. prove that X is a sub-gaussian random vector no matter if coordinates are independent or dependent. It is ...
0
votes
0answers
16 views

How to create a normal distribution over a finite set of points knowing max and min value of the distribution?

I need to build a model for camera lens deformation for a project I'm working on right now. Do do this, I'm trying to get a gaussian distribution of multiplicative coefficients over a finite array, ...
1
vote
1answer
30 views

Find likelihood for a given posterior

Let $p(z)=\mathcal{N}(z|0,1)$ be the standard normal density. Is there a conditional density $p(x|z)$ that could be expressed analytically and that would make the posterior $p(z|x)=\mathcal{N}(z|\phi(...
2
votes
1answer
18 views

What is the expectation of the product of a normalized (complex) gaussian vector and its hermitian transpose?

I have a complex Gaussian vector $\mathbf{x} \triangleq (x_1; x_2; ...; x_n) \in \mathbb{C}^{n \times 1}$, where $x_i \sim \mathbb{CN}(0,1)$ i.i.d., and $\mathbf{y} \triangleq (y_1;y_2;..;y_n)\in \...
0
votes
0answers
12 views

Z tables normal distribution $P(10 - c < X < 10 + c) = 0.95$. Find c. For N(10, 2)

$P(10 - c < X < 10 + c) = 0.95$. Find c attempt: $P(10 - c < X < 10 + c) = \phi(\frac{10 + c - 10}{\sqrt{2}}) - \phi(\frac{10-c-10}{\sqrt{2}}) = \phi(\frac{c}{\sqrt{2}}) - (1 - \phi(\...
0
votes
0answers
23 views

Solving equation concerning CDF of Normal random variable

Let $Y \sim \mathcal{N}(\mu,\sigma^2)$ and $F_Y(y)$ denote its cummulative distribution function. Any hint to solve the equation below for unknown $x$, known constant $a,b,c$? $$1-F_Y\left(a\sqrt{x}+...
1
vote
2answers
25 views

Understanding why variance of the standard normal distribution equals one intuitively

Can anyone explain to me why the variance of the standard normal distribution is 1? I am trying to understand the mechanism behind standardising random variable. While I know minus the variable by the ...
0
votes
0answers
10 views

Compute probability of a particular ordering of Multivariate normal distribution

There are $m$ normally distributed, independent random variables $N_0,...,N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
0
votes
2answers
25 views

Hypothesis testing using the 95% Confidence Interval of Sample Mean

I have a population mean of 120 (mu). I have a sample distribution with a mean of 131.05 and a standard-deviation of 11.00945. I have a sample size of 20, 19 degrees of freedom (n-1). I am performing ...
2
votes
1answer
32 views

characteristic function of $WX$ where $W$ and $X$ are independent standard normal random variables

Let $W$ and $X$ be independent random variables, both standard normal distributed. I have to show that for the characteristic function of $WX$ it holds that $\phi_{WX}(u) = \frac {1}{\sqrt{1+u^2}}...
0
votes
0answers
29 views

Derivative of CDF of Multivariate (Normal) Distribution

I am trying to get derivative of some probability functions and I do not know if it would is correct to use the following method. As an example, I am interested in derivative of $f(Q)$ with respect to ...
1
vote
0answers
17 views

How would one solve for a process where a stochastic random variable is divided by a deterministic random variable?

I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its ...
1
vote
1answer
25 views

Using the normal distribution to find the probability of getting an exact value

Take the following example. In the US, the average IQ is 100 points with a standard deviation of approx. 18 points. Find the following probability: Find the probability that a randomly selected ...
-3
votes
0answers
23 views

Derivative of the Inverse Cumulative Distribution Function($F(b*F^{-1}(x))$)

I tried to find the second derivative of $F(F^{-1}(x))$ where $F$ is standard normal CDF but I have difficulty hence please can anyone give me some idea? Thank you so much.
0
votes
1answer
40 views

Finding $Var(X)$ from conditional PDF

Let $X$ and $Y$ be random variables such that $X \vert Y=y$ is normal distributed as $N(y,1)$ and Y is a continues random variable with PDF: $f_Y(y)=3y^2$ for $0<y<1$ and $0$ otherwise. ...
-1
votes
1answer
37 views

Analytic result of the expectation of a quadratic under a Gaussian

In "http://rail.eecs.berkeley.edu/deeprlcourse/static/slides/lec-10.pdf", the instructor says that LQR works in stochastic dynamics in the same way as it works in deterministic dynamics, and mentions ...
1
vote
0answers
26 views

Domain of Central Limit Theorem

The central limit theorem says that if you take infinite number of samples ( > 30) from a population, compute their mean values, and collect them, you will reach normal distribution. Is this valid for ...
0
votes
1answer
10 views

How to divide a standard distribution in 5 groups?

For work I need to estimate how a group of 160 people would be divided among 5 sub-groups. Let's assume that the group conforms to a normal distribution. I need to split the group by expected output (...
0
votes
1answer
8 views

variance of conditional multivariate gaussian

I was playing around with Gaussian Distributions on my machine and I was interesting in making a pretty plot. I wanted to show the distribution of $x_1$ if $x_2$ was given if $x_1,x_2$ were ...
2
votes
1answer
19 views

Constructing a bivariate normal from three univariate normals

I'm trying to construct correlated bivariate normal random variables from three univariate normal random variables. I realize there is a formula for constructing a bivariate normal random variable ...
1
vote
0answers
19 views

Concentration of norm

Let $(X_1,...,X_n)\in \mathbb{R}^n$ be a random vector with independent sub-gaussian coordinates $X_i$ that satisfy $\mathbb{E}X_i^2=1$. Then $$||||X||_2-\sqrt{n}||_{\psi_2}\leq CK^2$$, Where $K=max|...
0
votes
1answer
23 views

Range of Data Values That Includes 95% of my Data

For a math assignment, I am required to find the range of data values that would include 95% of my data. My data is the wait time for a specific bus. I have gathered 20 pieces of data for this, ...
1
vote
2answers
45 views

Proving $(X,Y)$ is a normal vector when $X\sim N(1,1)$ and $Y\mid X\sim N(3X,4)$

Suppose I have a random vector $(X,Y)$ with $X\sim\mathcal{N}(1,1)$ and $Y|X = x \sim\mathcal{N}(3x,4)$. I need to prove that $(X,Y)$ is a normal vector as well. To do that I want to explicitly ...
1
vote
1answer
32 views

Can You Have 2 Numerical, Identical Values To Represent the nth Percentile?

For an assignment, I am required to find the value of a piece of data in the 70th percentile using mean and standard deviation. Here is my organized data in ascending order: ...
0
votes
1answer
30 views

Using change of variables to transform density functions

I'm was working on some exercises on statistical inference and came across a question I could not solve. After a while I decided to take a look at the solution to hopefully understand the problem ...
1
vote
1answer
31 views

Empirical distribution of sorted Gaussian numbers

I wrote a small program that does the following : Pick $N$ independent standard Gaussian numbers (expected value : 0, standard deviation : 1). Call that list $L=\{y_1, \ldots, y_N\}$. Sort that list ...
0
votes
0answers
11 views

Normalization for mixing cards

Let's assume I have N pair of king and queen cards. Each pair of king and queen was came together. What I want to do is to mix them together. Let's assume : $N^+ = No \ \ of \ King $ , $N^- = ...
1
vote
0answers
19 views

Closed-form solution for the expected value of the maximum of three independent normal variables

Is it possible to get a closed-form solution for the expected value of the maximum of three normally distributed variables that are independent but NOT identically distributed? In particular, I am ...