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

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Mean, variance and normal distribution

In a game of bridge hands of size 13 are dealt to each of 4 players in such a way that each hand can be considered to be a random sample without replacement from a standard pack of 52 cards. Each ...
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0answers
22 views

Coupling a chi-square to a normal random variable

Let $Z\sim \chi^2(k)$ be a random variable sampled from the Chi-Squared distribution with $k$ degrees of freedom. Vague question: Conditional on the value of $Z$, how can I reconstruct a sequence of ...
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1answer
17 views

Confidence ellipse for a 2D gaussian

For a 1D gaussian, the interval +/- 1SD about the mean will comprise ~68% of the area under the curve. Consider a 2D gaussian with a mean of zero and a diagonal covariance matrix (i.e., it is not ...
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1answer
11 views

The value of z representing the first Quartile of the standard normal distribution is:

I'm in desperate need of a hint at how they got the answer.
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1answer
29 views

Normal Approximation to the Binomial (Multiple Choice Question)

My first instinct in this question is use Normal approximation because N is large, and P is exactly between 1 and 0. I used the normal approximation, calculated when $p(X\le 19)$ and got 0.8997. The ...
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4answers
59 views

If $X \sim N(\mu, \sigma ^2)$, show that $(X - \mu) / \sigma \sim N(0,1)$ [closed]

I don't know how to do this. Do I need to use converge in distribution? (I thought this can only been used if $n$ involves)
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1answer
19 views

Argument shift Normal Distribution

In a mathematic book I have read following exercise: We throw a normal coin 10,000 times. The random variable $X$ tells us the number of tails. Give an approximation for $\mathbb{P}(4900 \leq X ...
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0answers
42 views

what is the expectation of $\sqrt{\left | x \right |} * sign(x)$ and $log(|x|)$ for a normal distribution

(1) What would $\int_{-\infty }^{\infty} \frac{\sqrt{\left | x \right |} * sign(x)}{\sqrt{2\pi}\sigma}e^{-0.5*\left ( \frac{x-\mu}{\sigma} \right )^{2}}dx$ evaluate to? This is expectation of ...
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1answer
31 views

Standard normal distribution inequality

I want to know how to prove the following inequality that seems to be true numerically. Let $n(x)$ be the density of the standard normal, and $N(x)$ be the cdf of standard normal. Then, for $x\geq ...
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0answers
21 views

Converting one normal distribution to another

I have a long data similar to this; between 15 and 25 consider its mean as m and calculated standard deviation using this formula. I assumed the above data as a normally distributed data and can ...
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0answers
21 views

Normal Distribution and optimization

Suppose the radius $X$ (in mm) of certain kind of water pipes follows the normal distribution $N(\mu,1)$. If the radius is less than 10 or larger than 12, then it is failed product. Suppose the ...
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1answer
17 views

Normal approximation to Binomial probability distribution

Where did this 0.5 come from? I understand we are using Z-score but in my calculations I basically omit the 0.5 to get a probability of .9616.
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2answers
51 views

Equation with normal distribution function

I was working on a task in probability, and got stuck at this: $ϕ(\frac{x-50}{4}) - ϕ(\frac{-x-50}{4}) = 0.6$ ($ϕ$ is the normal distribution function.) It's so simple, yet I don't know what to do ...
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1answer
35 views

normal distribution strange probability

Given the particular normal distribution specified below, what is the probability that a random observation falls within the specified range .004 greater and less than the average? original Lower ...
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1answer
25 views

How to calculate covariance of X and Y given joint probability

$X$ and $Y$ are dependent variables both normally distributed as $N(\mu-const, \sigma^2)$. I don't know what the joint distribution is. I know that when $const = 0$, then the joint probability ...
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3answers
50 views

Normal distribution exercise!

If a technician does not encounters any hardware problems, the time he requires to assemble a computer follows a normal distribution with a mean of $30$ minutes and a standard deviation of $3$ ...
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0answers
15 views

Normal distribution problem using R

The elongation of a steel bar under a particular load has been established to be normally distributed with a mean of $\mu = 0.05$ and a standard deviation of $ \sigma = 0.01$. Find the probability ...
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2answers
38 views

How to calculate the joint probability from two normal distributions

I have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution). $X$ and $Y$ are dependent. They are defined from other random variables A, B ...
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1answer
14 views

Optimize distributions for low mean, high variance

Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. ...
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1answer
42 views

Statistics - Exponential distribution

There are $n$ machines. Each has durability given by exponential distribution with $EX = 10$. If a dead machine is replaced with new one immediately, find minimal $n$ so we can say with $P = 0.99$ ...
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0answers
43 views

Standard normal distribution hazard rate

Is the hazard rate of the standard normal distribution convex? Can you give a reference?
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1answer
40 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
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2answers
29 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
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1answer
42 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
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1answer
23 views

Cholesky decomposition and variance

Well, I've been reading about simulating correlated data and I've come across Cholesky decomposition. Everything seemed clear until I found a couple of posts on this site and Cross-Validated that ...
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0answers
21 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
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3answers
100 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
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1answer
67 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
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1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
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1answer
16 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
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4answers
56 views

The normal distribution is a common model of randomness

Can someone please comment/elaborate on the statement: "The normal distribution is a common model of randomness." I would like to understand it more deeply. Source: Perhaps someone can point me ...
4
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1answer
32 views

Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
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1answer
37 views

Statistics- Finding Probability

A local lawn service has determined the average time it takes to mow an average residential yard is thirty-five minutes. If mowing times are independent and constant, what is the probability it will ...
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1answer
84 views

Checking the Lindeberg condition (central limit theorem)

Problem. Let $W_1, W_2,...$ be independent and identically distributed random variables such that $E(W_1)=0$ and $\sigma^2 := V(W_1) \in (0,\infty)$. Let $T_n = \frac{1}{\sqrt{n}} \sum_{j=1}^n a_j ...
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2answers
30 views

Gaussian function constant

Why are 1D gaussians defined as $$F(x;\sigma^2) = e^{\frac{-x^2}{2\sigma^2}}$$ for a probability function (after computing the gaussian integral): $$p_F(x;\sigma^2) = ...
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1answer
28 views

Probability of agreeing to do some work depending on the payment

I am looking for several options of modeling the probability of people agreeing to do some work depending on the price/payment. The payment can only range between $p_1$ and $p_2$, $(p_1 < p_2)$. I ...
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1answer
31 views

Question regarding the derivation of the distribution of $(n-1)S^2/(\sigma^2)$

I will quote from my statistics manual: "Consider a random sample $x_1,x_2...x_n$ taken from a population with distribution $N(\mu,\sigma^2)$, whose average $\mu$ is unknown; [through the central ...
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1answer
49 views

Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
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1answer
20 views

Expectation of normal and log normal distribution

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$. How do I find $$E[Xe^Y]$$? I tried a bunch of things without result. I'm also interested in "general" ...
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0answers
12 views

pseudo-Wishart distribution with shifted rows

I have a problem and I don't know where to start finding a solution. The problem is that I have a vector of i.i.d normal random variables such that, ...
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1answer
22 views

Properties of a distribution function

I'm having trouble understanding the properties of a distribution function. My book only gives these short rules. http://www.pixhost.org/show/2720/28297379_2015-06-22-15-27-44.jpg My professor said ...
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1answer
39 views

Correlation coefficients of X and Y [closed]

I was wandering if anybody could help me with the following question. I am fairly new to correlation coefficients and was attempting to tackle this question but was unsure how to do so? Thanks.
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1answer
51 views

To find $\sigma$ of a normal distribution

Given $X \sim \mathcal{N}( n, \sigma^2)$. The question told me $\mathbb{P}(X\lt 3) = \mathbb{P}(X\gt 7)$ So I found $n$ which is $5$. I'm also given $2\mathbb{P}(X\lt 2) = \mathbb{P}(X\lt 8)$. ...
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2answers
41 views

Explaining the form of the Gaussian measure

The Gaussian density $\mu(dx)=e^{-x^2/2}\ dx$ is fundamental in probability theory. Does anyone have a (non-computational) heuristic why this function should be special? (By non-computational, I mean ...
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0answers
23 views

How do I add uncertainties?

I have a gas for which we are continuously measuring composition with an online instrument. This composition is then being used to calculate some properties (dewpoint), and I want to estimate the ...
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10 views

Dice probability normal distribution

You roll a dice 1000 times. Calculate the probability you roll a six between 150 and 200 times. I understand how you calculate this with the binomial distribution: $$ = Binomialcdf(1000, 1/6, 200) - ...
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1answer
120 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
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1answer
23 views

A convergent sequence of normal random variables

Say $\{X_n\}$ is a sequence of normal random variables with means $0$ and variances $\sigma_n^2$. Also suppose that $X_n\to X$ (everywhere) and $\sigma_n^2\to \sigma^2.$ Then, using characteristic ...
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0answers
31 views

Equally spaced numbers with interval

I've this very silly doubt in getting equally spaced numbers with fixed interval from a set of numbers. Let's say I have a set of points, if I want maximum N points from this set equally distanced, ...
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0answers
35 views

Probability involving Normal Distribution

Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056h. The error made by the ...