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

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0
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1answer
22 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.
0
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1answer
37 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 ...
0
votes
4answers
150 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)
1
vote
1answer
34 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 ...
2
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0answers
52 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 ...
0
votes
1answer
40 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 ...
0
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0answers
78 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 ...
0
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0answers
30 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 ...
1
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1answer
22 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.
0
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2answers
55 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 ...
0
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1answer
40 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 ...
0
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1answer
68 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 ...
1
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3answers
87 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 $3$ ...
0
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0answers
69 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 ...
0
votes
2answers
131 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 ...
0
votes
1answer
22 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. ...
1
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1answer
52 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 $P = 0.99$ ...
1
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1answer
279 views

Standard normal distribution hazard rate

Is the hazard rate of the standard normal distribution convex? Can you give a reference?
-1
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1answer
46 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 ...
0
votes
2answers
95 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 ...
2
votes
1answer
46 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 ...
2
votes
1answer
55 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 ...
1
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0answers
30 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 - ...
2
votes
3answers
131 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 ...
5
votes
1answer
68 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 ...
3
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1answer
86 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 ...
0
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1answer
37 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 ...
1
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4answers
152 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
votes
1answer
93 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 ...
1
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1answer
39 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 ...
5
votes
1answer
227 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 ...
2
votes
2answers
42 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) = ...
0
votes
1answer
32 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 ...
1
vote
1answer
36 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 ...
2
votes
1answer
56 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} ...
0
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1answer
30 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" ...
0
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0answers
19 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, ...
0
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1answer
30 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 ...
0
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1answer
44 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.
1
vote
1answer
53 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 $2\mathbb{P}(X\lt 2) = \mathbb{P}(X\lt 8)$. ...
0
votes
2answers
50 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 ...
0
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0answers
27 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 ...
1
vote
0answers
28 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) - ...
6
votes
1answer
237 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 ...
0
votes
1answer
34 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 ...
0
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0answers
69 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, ...
2
votes
2answers
141 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...
0
votes
1answer
49 views

How to calculate $E(\sin^2X)$

If $X \sim N(0,1)$ then calculate $E(\sin^2X)$ I understand that $0 < \sin^2x<1$. So the expectation exists. I proceed as $E(\sin^2X)= \int_{-\infty}^{\infty}\sin^2xf(x)\,dx=2 ...
0
votes
1answer
79 views

How to calculate expected value of normal distribution with the condition that value is higher than x

I have following problem. Let assume that lifespan in the population has normal distribution with certain mean, variance and skewness. When the baby is born, its average lifespan will be equal to ...
0
votes
1answer
33 views

Show that for smaller $n$ than $n = 125000$ holds: $\mathbb{P}(|Z_n - \frac{1}{2}| \geqslant 0,01) \leqslant 0,02$

I'm a first year math student and I am having trouble with this exercise: Let $S_n$ be the amount of times we get heads when throwing a coin $n$ times. Let $Z_n = \frac{S_n}{n}$. With the equality ...