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

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Rearrange for solving x (Miller & Siegmund, 1982; equation 8)

I have the following formula from a paper back in 1982 by Miller & Siegmund, "Maximally Selected Chi Square Statistics": α = 0.05 φ() is the standard normal density function: Everything else ...
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2answers
107 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
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1answer
851 views

How can you normalize two data sets to the same scale?

I have two data sets, one that ranges from 0-200, and another that ranges from ~400-~2500. I would like to compare the two according to a score from 0-10. I remember about normalizing from a ...
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0answers
28 views

Non-Geometric Proof of Random Normal Projection Identity

Many papers on locality sensitive hashing, sketching and similar use the following lemma: If $r\in\mathbb{R}^d$ is a random vector with all entries normally, independently distributed as ...
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14 views

central limit theorem and sampling dist.

If you takes samples from a distribution, and you can see that they have different variances, can the central limit theorem still be applied. The computer vision teqnique i am referring to is this ...
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2answers
74 views

Robust estimator

What does it mean that an estimator is robust? How can you tell whether an estimator is robust or not in statistics? I need to discuss whether the maximum likelihood estimators of the normal ...
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1answer
61 views

A question on a certain transformation of a normal random variable

I have to solve the following exercise: Suppose $X\sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi,\Phi$ are the pdf and cdf of the standard normal distribution with ...
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1answer
43 views

Intuition for proof of Slepians Inequality

If z is a centered gaussian random variable and $ x_1 ,x_2 ,..,x_n ,y_1,y_2,..,y_n $ are points in $ \mathbb{R}^{2n} $ satisfying $ |x_i-x_j |_2 \leq |y_i -y_j |_2 \ \ \ \forall i,j \in [n] $ then $ E ...
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56 views

Show Almost Certain Convergence of a Sequence of Normal Random Variables

Let $(X_n)_{n=1}^\infty$ be independent, $N(0,1)$-distributed random variables. Prove that $$ \limsup_{n \to\infty}{X_n \over \sqrt{2 \log(n)}} = 1 \ \text{almost surely}.$$ I am aware of the ...
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1answer
125 views

Integral of normal distribution curve

I am having hoping to use the integral of the normal distribution curve to find the probability of having a mean of $0.30$ or greater, i.e. one tailed distribution. With a sample standard deviation of ...
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1answer
58 views

Standard Normal Distribution Transformation Z=lnY

I'm not sure if my approach to this problem is correct and I need help I need to apply $Z=\ln{Y}$ to the following standard normal distribution and then find the distribution of $Y$ ...
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1answer
56 views

Sum of weighted normal distributions, how to solve $P(X<x) = y$ for $x$?

How do I solve the following equation for $x$ ...
2
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1answer
504 views

Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution

I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was ...
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2answers
42 views

Probability that $x > 0$ given that $x > y$ for independent and normally distributed $x,y$

I was recently been asked this question in an interview but not able to solve it as I am rusted in Bayesian conditional probability. Here is the question: $x$ and $y$ are independent variables ...
2
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1answer
43 views

Why is the $0$th percentile of the standard normal distribution $-\infty$?

Why is the $0$th percentile of the standard normal distribution $-\infty$? I can't explain the cause except saying there is no area under the curve. So it goes beyond the bell-shaped curve.
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1answer
59 views

Independence of Gaussian Distribution Function with Different Means

Is there any way to prove that $N$ Gaussian distribution functions are linearly independent if and only if the means are different. For example, if ...
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1answer
180 views

How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
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2answers
29 views

How to analytically find probability of certain combinations between normally distributed populations?

Let's say we have two normally distributed populations A and B, which have different means and standard deviations. We then pick one item from population A and one from population B. How could we ...
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1answer
37 views

Is the following process bounded (iterative normal sampling)

We define the following stochastic process: $X_0=1$ $\forall i\geq i:X_i\sim\mathcal N(0,X_{i-1}^2)$ That is, we first sample $X_1$ from the normal distribution with variance $1$, then in the ...
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0answers
26 views

Effect of Normalization On a Distribution

Let's assume that we have calculated some values proportional to probabilities for a special distribution(some hypothetical distribution other than normal distribution but we don't know which ...
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1answer
100 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
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1answer
43 views

$P(X=c)=0$ for normally distributed $X$?

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
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1answer
83 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
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1answer
65 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density surface of a Gaussian mixture distribution. Is such an iso-density surface a union of ellispoids? Let's say that this Gaussian mixture ...
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1answer
489 views

What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
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1answer
31 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
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2answers
1k views

Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
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1answer
104 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
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1answer
467 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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1answer
106 views

When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
2
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1answer
38 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
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2answers
60 views

How to show that normal random variables U1 and U2 are independent?

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
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1answer
122 views

Conditional probability for two normal distributed variables.

I haven't had to do much with probabilities since university, so please excuse if this is trivial or the question is not well specified. Let $X$ and $Y$ be two independent, normally distributed ...
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1answer
58 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
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1answer
24 views

Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... ...
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1answer
376 views

Word problem related to normal distribution

Consider the following problem: WINK, Inc. is made up of 450 employees who work a total of 13,500 hours per week. If the number of weekly work hours per person has a normal ...
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1answer
292 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
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1answer
3k views

Expected value of lognormal distribution.

Hi I'm stuck on this question: Recall that X is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if log(X) is normal with mean $\mu$ and variance $\sigma^2$. Suppose X is ...
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0answers
25 views

normal approximation to poisson

Cotton yarn is wound onto bobbins, each of which takes $100$m of yarn. If the thread breaks before $100$m is reached, the bobbin is rejected. In a trial of a new spinning machine, $13$ bobbins out of ...
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2answers
536 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & ...
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2answers
104 views

Evaluating Integral involving exp

I am stuck at the following integral :- $$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$$ Can anybody give me some hint. and also for this function $$ \int_{- ...
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1answer
59 views

Probability of an event occurring more than X% of the time

Cans are a normal random variable with a mean of 7.96 ounces and a standard deviation of 0.22 ounces. Suppose that you draw a random sample of 34 cans. Use normal approximation to find the ...
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1answer
38 views

Transforming a normal distribution to a uniform one

I'm searching for a method that transforms a normal distribution into a normal distribution. I've looked everywhere, but I'm not sure if I just missed something completely obvious, if this actually is ...
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0answers
26 views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
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2answers
125 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
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1answer
52 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
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2answers
150 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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1answer
65 views

Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
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1answer
27 views

Identity involving the relation Normal Distribution and Other arbritary Distribution

Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ ...
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2answers
68 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0 +$ integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.