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

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2answers
34 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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0answers
10 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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0answers
94 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
299 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
22 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
22 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
22 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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0answers
36 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 ...
2
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1answer
48 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 ...
9
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1answer
119 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 ...
2
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1answer
42 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
64 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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0answers
36 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 ...
2
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1answer
64 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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0answers
11 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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0answers
21 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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1answer
38 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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0answers
33 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
14 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
31 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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0answers
30 views

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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0answers
34 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
39 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
22 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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1answer
47 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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1answer
29 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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0answers
12 views

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
61 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
15 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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0answers
24 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 ...
2
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2answers
60 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
62 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 ...
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1answer
10 views

Converting normalised values into original

I have a normalisation formula as follows, which takes a list of numbers, such as $1,2,3,4,5,6,7,8,9,10$, and returns the normalized values which guarantees that $\tilde{x_i} \in [0,1]$. ...
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0answers
24 views

Sampling Distribution of the Mean

I want to know if my reasoning is correct. Let's say I got two normal distributed variables: Variable "X": 5.4 (mean), 2.856 (variance) Variable "Y": 5.4 (mean), 5.062 (variance) Let's pick 16 ...
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1answer
134 views

How do I plot normal distribution

If I know the range (1-24) and know the area (X), how can I plot a normal distribution so that the curve has area X?
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1answer
54 views

Total variation distance of two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$

I need to prove that the total variation distance between two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$ converges to $0$ when $s \nearrow t$. We know that ...
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1answer
64 views

Estimating the probability of a failure

We are estimating the spares requirement for a radar power supply. The power supply was designed with a mean ($\mu$) life of $6500$ hours. The standard deviation ($\sigma$) determined from testing is ...
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1answer
48 views

Prove that $Y = \frac{X_1+X_2*X_3}{\sqrt{1+X_1^2}}$ obeys normal distribution

given that $X_1, X_2, X_3$ are independent and identically distributed, $X_1 \sim N(0,1)$. I tried to calculate the cumulative distribution function of Y: \begin{align} P(Y\leq y) &= ...
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1answer
22 views

sum of iid variables: how many terms needed for convergent to normal

For sum of iid variables $Z_n=\sum_{i=1}^nX_i$, in general, how large should $n$ be to indicate 'convergence' to normal? 10? 100?
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1answer
41 views

Correlation coefficient and Expectation of two dimensional normal distribution.

Random variable (X,Y) is normally distributed. Conditional expectations are $E(X|Y=y)=0.25y + 2$ $E(Y|X=x)=x-2$ How can i determine correlation coefficient and when that is known, the expectations ...
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2answers
84 views

Approximation of uniform distribution.

There are leaving from the station arriving every 10 minutes. A person has to wait from 0 to 10 minutes at the station, this is uniformly distributed. Now if the person uses the station 100 times a ...
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1answer
36 views

finding the pdf for $y$

Let $X\sim N(0,1)$ that is $X$ is a random variable with normal distribution with mean$=0$ and standard deviation$=1$ and $$f_X(x)=\dfrac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$$ Let $y=g(X)=\dfrac{1}{x}$. ...
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0answers
18 views

Help relating gaussian to chi-squared distribution

I am having trouble finding a simple layout/documentation for the chi squared distribution. From what I understand the chi squared distribution is just: Where "v" is some strength parameter. Now, ...
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0answers
12 views

Computing CDF of Normal-Bernoulli

I have the following setting: Let $\theta \sim N(\mu,\sigma^2)$ and \begin{equation} e = \left\{ \begin{array}{l l} E & \quad \text{if} \quad \theta > \theta^* \\ 1 & \quad \text{if} ...
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3answers
37 views

Characteristic function of a square of normally distributed RV

Let's assume that $ X \sim \mathcal{N}(0,1) $. I'm supposed to compute the characteristic function of $ X^2 $. As far as I got is that the density of $ X^2 $ is $ g(y) = \frac{1}{2\sqrt{2\pi}} ...
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1answer
71 views

Characteristic function of a product of random variables

I am facing the following problem. Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $. I have found some information, ...
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1answer
50 views

Multiple hypothesis testing

Let's suppose I have 10 independent measurements with results close to zero. How can I claim that they are in agreement with the theory, them being zero? The errors of these 10 results are not equal, ...
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1answer
17 views

Is the linear transform of joint gaussian necessary gaussian? See this case!

Suppose we map the low dimensional Gaussian distribution into higher dimension using linear transform. Say, $X \in R^p$ is joint Gaussian, and for $n > p$, $Y = A_{n \times p}X$. Is $Y$ joint ...
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0answers
149 views

Using the central limit theorem to prove a statement regarding normal distribution, from a population with exponential distribution

X1, . . . , Xn are a random sample from a population having an exponential distribution with rate parameter λ. Use the Central Limit Theorem to show that, for large values of n, sqrt(n)*(λx − 1) ∼ ...
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3answers
115 views

Probability of two normal random variables when random samples are taken from a population

This is sort of second section to my previous question, I should have included both together, but I forgot to. Sorry for any inconvenience. X= random height of a male Y= random height of a female X ...