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

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20 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
16 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 ...
2
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0answers
12 views

Prove $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$ for $F$ is the standard normal CDF

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 one ...
2
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1answer
30 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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20 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
6 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
33 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
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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0answers
12 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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16 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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0answers
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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0answers
22 views

Normal Distribution(Probability) [on hold]

(a) The hourly wage of employees in a certain service industry is believed to follow the Normal Distribution N(40, 5^2) which has a mean µ of 40 dollars and a standard deviation σ of 5 dollars. The ...
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0answers
24 views

Poisson and Normal Distribution [on hold]

Hits to a high-volume Web site are assumed to follow a Poisson distribution with a mean of 10, 000 per day. (a) Approximate (by a normal distribution) the expected number of days in a year (365 days) ...
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0answers
13 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
10 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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40 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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0answers
17 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
7 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
24 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
25 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
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
27 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 ...
1
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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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0answers
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 ...
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0answers
13 views

Math Statitics. Normal Distrubuton [closed]

Studies shown that gasoline use or 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 ...
3
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1answer
30 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
36 views

Exponential to Normal approximation [closed]

Suppose that the length of life of a piece of equipment is exponentially distributed with a mean length of 30 days. As soon as it fails another is installed in its place. Find the probability that ...
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0answers
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 ...
1
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1answer
21 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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10 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
35 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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0answers
18 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
51 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
51 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
9 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
20 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
26 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?
5
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1answer
35 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
48 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
42 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
12 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
28 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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1answer
29 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
31 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
13 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
9 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
18 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
26 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
45 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, ...