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

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Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
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18 views

Empirical Rule. Is it applicable in this case?

So I ran in this problem: I have to test whether Empirical Rule is applicable. Proportions I got is 73%, 94,7% and 99.1% (within one, two and three standard deviations). I'm worried about 73%. This is ...
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18 views

Relative Error of Normal Approximation

I have this math statistics assignment that I worked out and came out with a couple of graphs. How do I interpret the graphs? Task: If you consider the average of 20 iid rvs, normal distribution is ...
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1answer
32 views

Student's distibution

If $X_i$ are independent equally distributed random variables, $S_n=X_1+...+X_n$, then $$\frac{S_n-n\mathbb E(X)}{\sqrt{n \sigma^2(X)}}$$ tends by distribution to $N(0,1)$ for $n \to \infty$. It is ...
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25 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
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20 views

$L^2$ limit of Gaussian random variables

Let $X_n$ be a sequence of Gaussian random variables defined on the same probability space. The statement is that if $X_n$ converges to some random variable $X$ in $L^2$-sense, then $X$ is also ...
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1answer
32 views

Exponential distributed with expected value

Have this question in math statistics Normal Distribution. In a certain cellular phone system new calls arrives with exponential distributed interarrivaltimes with expectation value $$\mu ...
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38 views

Let $Y_1, Y_2, … Y_n$ be a random sample from a normal distribution…

Homework problem here. The teacher didn't exactly go over this stuff, so I just want to make sure I have this correct. Let $Y_1, Y_2, ... Y_n$ be a random sample from a normal distribution where the ...
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1answer
34 views

It is estimated that 80% of all eighteen-year-old women…

Homework question here, I'm not sure how to finish it out. It is estimated that $80\%$ of all 18-year-old women have weights ranging from 103.5 to 144.5 lb. Assuming the weight distribution can be ...
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34 views

Reasons for data to be normally distributed [migrated]

What are some theorems which might explain (i.e. generatively) why real-world data might be expected to be normally distributed? There are two that I know of 1) The Central Limit Theorem (of ...
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8 views

Projection of a bidimensional normal distribution on a plane

I have a bivariate normal distribution with zero mean and the same $\sigma$ on the two axes: ...
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2answers
50 views

Sum of 'inverse' Normal (1/X) random variables. Equivalent resistance calculation

Consider the case of $N$ resistances $R$ connected in parallel. The equivalent resistance of such a circuit is calculated as follows $$ \frac{1}{R_{eq}} = \underbrace{\frac{1}{R} + \frac{1}{R} + ...
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2answers
27 views

Find area under the curve of a standard normal distribution

Given a standard normal distribution, how can I find the area under the curve that lies: 1. to the left of z = −1.39; 2. to the right of z = 1.96; 3. between z = −2.16 and z = −0.65;
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8 views

Normal distribution calculations

i) Y~N$(3.5,1)$, P$(|Y - 3.5| > 1.5)$ ii) X~N$(3,1)$, Y~N$(5,2)$, X,Y independent; P$($Y $< 2X)$ Not sure on how to calculate these distributions.
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38 views

Uniform distribution in a cube

I came across the following problem and got stuck. Problem: Let $X_1,X_2,...$ be independent Unif$(-1,1)$ and $S_n=X_1^2+...+X_n^2$. Let $$A_n=\{x\in ...
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1answer
53 views

create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...
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1answer
15 views

Distribution of sample variance from normal distribution

Assuming $N$ samples $\{x_1,...,x_N\}$ are taken from a normal distribution with mean $\mu$ and variance $\sigma^2$, then the variance can be estimated using \begin{equation} ...
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0answers
11 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
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1answer
19 views

Determining a conditional probability with a random variable.

Assume $X$ is a normal distributed random variable with mean $2$ and variance $4$. Determine the conditional probability $P(1 \le X \le 3|0 \le X \le 4)$ What I did: $$Z_0 = \frac{0-2}{2}=-1$$ $$Z_1 ...
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34 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
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1answer
41 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
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1answer
109 views

Inequality for $N(0,1)$ CDF: $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does ...
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1answer
38 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the ...
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1answer
51 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
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12 views

Find the UMP test for Hypothesis testing

Let $X_1,...,X_n$ be a random sample from the $N(μ,σ^2)$. Assume $μ=0$ (a) Find the UMP test for $H_0 : σ^2=σ_0^2$ versus $H1 : σ^2 < σ_0^2$ at signicant level $\alpha$. (b) For the UMP test in ...
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1answer
42 views

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
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5 views

Asymptotic confidence interval

Let x1, x2, ..., xn be a random sample with a density function given by $ f(x) = \frac{3}{\theta^3} x^2 I_{(0,\theta]}(x)$ where $I_{(0,\theta)}(x)$ is the indicator function and $\theta > 0$ ...
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14 views

How to check $H_0$ hypothesis using Pearson's criteria?

How to check hypothesis by using Pearson's criteria ( $\chi^2$ test), that $H_0:$ random variable $X$ is normally distributed given that $k=7$ (count of intervals) and $\alpha=0.1 $ (significance ...
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20 views

Statistics and mathematics

fifty five percent of the registered voters in sheridan ville favor their incumbent mayor in her bid for re-election. If four hundred voters go to the polls, approximate the probability that the race ...
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52 views

Marginalizing product of multivariate normal distributions [migrated]

How should I marginalize $F_{i}$ from the following probability distribution $$p(y_{i}|F_{i},\alpha, \Lambda, \Phi, \Sigma) = N(\alpha + \Lambda F_{i}, \Sigma)$$ in order to obtain ...
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16 views

How to check null hypothesis in Minitab without specific data?

I have been given a question that specifies a sample size of 50, sample mean of 3.05, standard deviation of .34, and desired mean of 3.2. The question asks whether or not the average mean is ...
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1answer
12 views

relationship between two normally distributed variables

Say I have two normally distributed independent random variables (X1 and X2) with the same variance but different means. How would I calculate P(X1 > X2)?
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41 views

Multivariate Normal Variable (Homework)

I am trying to solve the following question: Let $(V, Z) ∼ MVN(0,I)$ (where I is the identity matrix) and let $Y=V+Z+1$. Find the distribution of $(1+Z,1-Y)$. I have found the distribution of ...
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20 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
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2answers
9 views

normal distribution question with percentages

how a can i solve a normal distribution without the mean ? suppose a truck of river sand delivered by a company has normal distribution with a standard deviationof 100kg.if 20% of loads are at least ...
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1answer
25 views

Distribution for random variable Z = Y1 - Y2

This was one of the interview questions. I did not know the answer. Question : Let Y1 and Y2 be two independent random variables where Y1 follows Normalpdf[x, -2, 5] distribution and Y2 follows ...
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What is $E[\cos X]$ where $X$ is lognormal?

I was asked in an interview to compute $E[\cos X]$ where $X$ is lognormal. I tried using lognormal's characteristic function (Taylor series representation, which is divergent) and $\cos ...
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29 views

How to calculate this kind of probability for a normal distribution?

here is my question. I have a normal distribution with known mean and variance. Say the mean is 3 and the Var. is 2. what is the probability that the random variable is taking value 2.9? If I plug ...
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1answer
28 views

When to expect normal distribution?

I was wondering when a normal distribution can be expected. I know that things like: heights of people size of things produced by machines errors in measurements blood pressure marks on a test ...
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2answers
18 views

Studies shown that gasoline use for compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg.

Studies shown that gasoline use for compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg. Find the range of mileage for the middle 60% of ...
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22 views

How to generate random normal skewed distribution?

We have right skewed normal distribution dataset whose mean is ~180, SD is ~60 and Skewness is 1.64. We have calculated Skewness using skewness function of R package "e1071" How do we generate ...
3
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1answer
32 views

Find the cutoff level for the highest 15% in normal distribution, given the mean and standard deviation [closed]

The cholesterol levels of adult American women are approximately normal with the mean of 188 mg/dl and a standard deviation of 24 mg/dl. a company wants to test a certain medication for women ...
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25 views

Airplane Overbooking Problem

Sometimes customers will make a reservation and then not turn up. To off-set this problem some companies may decide to “overbook” so they are not left with empty places. For example, an airline ...
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1answer
25 views

Applying a Normal Distribution to Another Function to Find Probability

Suppose that the number of hours students spend studying for an exam is approximately normally distributed with $\mu=10$ and $\sigma=\sqrt{2}$. If a student spends $t$ hours studying, he/she ...
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Suppose the number of hours that a student spends working on an assignment is approximately normally distributed…

with mean $\mu = 10$ and variance = 2. If a student spends t hours working on the assignment she receives a mark of M(t): $M(t) = \frac1{1 + e^{-t+7} }$ What is the probability she receives at least ...
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1answer
39 views

Box-Muller Transformation

I know that we can use the Box-Muller transformation to generate a pair of independent standard Gaussian random variables using a pair of independent standard uniform random variables. I am wondering ...
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1answer
7 views

How can I find the percentile function of a distribution that isn't normal?

I know that: $$ X = \mu + Z\sigma$$ for a normal distribution. I'm having a tough time understanding where this is derived from, though. How is it found and how is it found for other distributions?
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23 views

Transformations of Normal Distribution

Let $X \sim \mathcal{N}(0, 1)$. We define the CDF, $\Phi(x)$, of $X$ as: $$ \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-\frac{t^2}{2}}\,\mathrm{d}t $$ If $Y=\Phi(X)$, what is ...
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2answers
59 views

You purchased stock for \$1m. What is the probability that it is worth more than $30m after 10 years?

The change in value of the investment each year is modeled as follows: Divided by 2: 1/4 Remain unchanged: 3/8 Doubles: 1/4 Quadruples: 1/8 Where I'm at: I'm aware that this needs to be formulated ...
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2answers
53 views

Let X be normally distributed with mean $0$ and variance $1$, find the CDF and density of $Y = \Phi(X)$

Define $\Phi(x)$ as: $$ \Phi(x) = \frac 1{\sqrt{2\pi}}\int_{-\infty}^x \exp\left(-\frac{t^2}{2}\right) dt $$ and let the random variable $Y$ be defined as $\Phi(X)$ where $X$ is a standard ...