0
votes
1answer
7 views

Modelling a normal-like single-ended random variable

I am trying to model a of (normal-distribution-like) discrete random variable using the normal distribution. This is what I understand so far: First, I approximate the mean of the normal ...
0
votes
1answer
48 views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
2
votes
1answer
24 views

Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
0
votes
1answer
21 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
1
vote
1answer
67 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
0
votes
1answer
28 views

Stats Normal distribution [closed]

A company pays it employees an average wage of $\$15.90$ an hour with standard deviation $\$1.50$ per hour. Assume the wages are approximately normally distributed. a) What proportion of employees ...
-2
votes
0answers
11 views

normal dribution -with mean time and standard deviation [closed]

The time taken by passengers arriving at an airport has been recorded as follows: The mean walk time from the chocks on of an aircraft that lands to the immigration counter is 5 mts, with a standard ...
0
votes
2answers
26 views

Area under Normal Distribution Curve

What is the formula that determines the Z-score table? More specifically, what formula can be used the equate the area underneath the normal distribution curve, without using the table?
0
votes
1answer
24 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
-4
votes
1answer
81 views

Prove that if $X \sim N(\mu, \sigma^2)$, then $X \sim \mu + \sigma N(0, 1)$ [closed]

As above. Also how is the general case proved for multivariate Gaussian? edit: I'm not sure why people voted to put this on-hold, it's just asking for a justification of a commonly used statistical ...
0
votes
1answer
32 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
0
votes
3answers
42 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
0
votes
1answer
19 views

Plotting Normal Distribution using Excel

I was trying to experiment some stuff (scaling issues and hypothesis testing) with normal distribution. While doing so, I found out that : NORM.S.DIST(0, FALSE), which takes Z-value, returns prob. ...
1
vote
1answer
11 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
1
vote
0answers
16 views

Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
1
vote
1answer
52 views

Determining whether random variables are independent

If I have two random variables as follows: 1) A Gaussian distribution of wifi signal strengths at a known point 2) A Gaussian distribution of wifi signal strengths at an unknown point (Note that ...
0
votes
2answers
22 views

Calculating Variance

Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a population whose distribution is normal with mean $\mu$ and variance $\sigma^2$. Consider the statistics $\displaystyle T_1 = \frac{X_1 − X_2 ...
1
vote
1answer
31 views

Estimate variance, how to find expected value of $x^2 [n]$

We have data $x_0, x_1, \ldots, x_{N-1}$ where the $x_n$'s are independent and identically distributed as ${\rm Normal}(0,\sigma^2)$. The estimate of $\sigma^2$ is $$\hat \sigma^2 = \frac{1}{N} ...
1
vote
1answer
57 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
-4
votes
1answer
39 views

If a random variable X has mean of μ and standard deviation σ…

, then what will be the mean and standard deviation of (X − μ)/σ ?
2
votes
1answer
52 views

if $X_i$ are iid standard normal distributed, what is the limiting distribution of $\sum X^4 / (\sum X^2)^2$?

If $X_i$, $i=1,\ldots,n$ are iid standard normal distributed, what is the limiting distribution of $S_n=\sum X^4 / (\sum X^2)^2$? After finding the moments and since $Cov(X^4, X^2)=0$, I have the ...
0
votes
2answers
34 views

Given N-distribution, calculate expected value and Var of a function

$X_{{i}}=X_{{1}}...X_{{n}}$ is an iid. random variable with the distribution $N \left( {\frac {\alpha}{\beta}},{\frac {{\alpha}^{4}}{{\beta}^{4}}} \right) $ I would like to calculate the expectation ...
1
vote
2answers
27 views

MLE of MVN($\mu, \Sigma$)

I'm trying to find MLE of MVN($\mu, \Sigma$), i.e $N_k(\mu, \Sigma)$ with random sample $X_i, 1\le i \le n$. It was easy to get $\widehat{\mu}= \bar{X}$ and $\hat{\Sigma} = \frac{1}{n} \sum_i (X_i - ...
3
votes
1answer
98 views

Linear combination of normally distributed variables

We know that if $X \sim N_p(\mu, \Sigma)$ then $a'X \sim N(a'\mu,a'\Sigma a)$ for and $a \in \mathbb{R}_p$. What I need to know is if the converse of this is also true. Can this be proved? Would ...
0
votes
2answers
39 views

“Show experimentally” that for large $N$, $X$ appears to be normally distributed.

I'm a bit confused about the following problem: Let $X$ be the random variable $$X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}$$ where $X_k$ is the outcome from the $kth$ flip of a fair coin where heads ...
0
votes
0answers
22 views

For which joint distributions is a conditional expectation an additive function?

I know that, for a random vector $(X,Y,Z)$ jointly normally distributed, the conditional expectation $\mathbb{E}[\,X\mid Y=y,Z=z]$ is an additive function of $y$ and $z$. For what other distributions ...
0
votes
1answer
18 views

Does correlation have to be in the context of (Gaussian) normal distribution?

I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as: $\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ...
0
votes
1answer
37 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
4
votes
1answer
130 views

Probability of a gaussian distribution in another gaussian distribution

Assume we have a Gaussian distribution $p(x) \sim \mathcal{N}(\mu_p,\Sigma_p)$ For any point $X$, it is easy to compute the density of $x$ in $p$: $$p(x) = \frac{1}{|2\pi ...
1
vote
0answers
38 views

How to use the normal probability table in reverse

I'm just wondering if anyone could give me a bit of advice on this. This relates to CCEA's S1 exam questions. $Z \sim \text{N}(0, 1)$ Let's say $\phi(z) = 0.5015$ Find z. Here is an extract of the ...
0
votes
0answers
30 views

Integral of multivariate normal density function

Is anybody know a suited close-form solution for this integral: $$ I=\int_{R^n} x_i \cdot x_j \cdot f_N({\bf x},{\bf \mu},{\bf \Sigma}) d{\bf x} $$ where ${\bf x}=\{x_1,\ldots,x_n\}$ and $f_N$ is the ...
5
votes
1answer
94 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
0
votes
0answers
12 views

Proving limit of variance estimator (Normal distribution)

i have a problem with a exercise from my statistics I book, some help would be appretiated ... Let $x_1,x_2,...,x_n$ random sample from normal distribution $N(\mu,\sigma)$, where $\mu$ and $\sigma$ ...
0
votes
1answer
28 views

P-P plot and Q-Q plot

How to draw P-P plot and Q-Q plot manually ? I have looked at different site and they explained in various way, such as one said for p-p plot in X-axis there is residual in ascending order and in ...
1
vote
1answer
21 views

limiting behavior of standard normal survivor function [duplicate]

How do you show that $\lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x$? In the previous, I'm using $\Phi$ to refer to the standard normal CDF and $\phi$ to refer to the standard normal pdf. Thanks!!
0
votes
1answer
40 views

Normalizing a dataset from the interval [0,1] with fixed properties.

So I have a rather large dataset where values are from the interval $[0,1] \in \mathbb{R}$. But the problem is that a big portion of the values are extremely close to $0$. So firstly I'm looking for ...
1
vote
3answers
28 views

Show that $Y\sim N(a+A\mu,AVA^T)$

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$. Show: If $a\in\mathbb{R}^m$ and $A$ is a $(m\times n)$-matrix with $\text{rang}(A)=m$ then $$ Y=a+AZ\sim N(a+A\mu,AVA^T). $$ My ...
0
votes
0answers
8 views

Uniqueness of zero point for a function related to normal cdf

Let $a,b,c$ be three given real numbers with $0<a<b<c$. $\Phi$ is the cdf for standard normal distribution. Let $A_1 = \Phi (x+a) - \Phi (x)$, $A_2 = \Phi (x+c) - \Phi (x+b)$, and $\delta(x) ...
1
vote
1answer
12 views

Normally distributed data or not

Can I say that the datas are normally distributed? I would say yes, but I am not entirely sure.
1
vote
1answer
24 views

A practical question in statistics

A student leaves home at 8 a.m. every morning in order to arrive at the University at 9 a.m. He finds that over a long period he is late once in forty times. ($\frac{1}{40}$) He then tries leaving ...
0
votes
1answer
46 views

Likelihood ratio critical region

Let $X_{1},..,X_{n}$ be a random sample from a normal distribution with mean ${\theta}$ and variance 1. We wish to test $H_{0}:{\theta}=0$ vs $H_{1}:{\theta}{\neq}0$. Write down likelihood ratio ...
1
vote
1answer
144 views

mean and variance normalization of vectors

I have vectors $x \in \mathbb{R}^n$ and I expect some multivariate normal distribution. I want to normalize the vectors in such a way that $y = M (x - b)$ has mean zero ($\operatorname{E}[Y] = 0$) ...
0
votes
1answer
39 views

Standard deviation with multiple means and deviations

The amounts of a certain mineral that can be produced in a day from mines $1$, $2$, and $3$ are independent normal random variables with means equal to $80$, $90$, and $75$ pounds, respectively, ...
1
vote
1answer
22 views

Exponential deviation with two $x$ values

I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ...
1
vote
0answers
19 views

Finding the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$

Suppose $X=[X_{1},X_{2}]$ and $X$~$N_2(μ,Σ)$. I wish to find the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$. Since this is of a quadratic form I do not know a way of solving this. However I kind ...
0
votes
0answers
24 views

Normal Distribution while finding sigma

I was reading some things about normal distribution and saw this problem in a text a couple days ago. I know it might be a little advanced for me at the moment, but I was wanted to know if someone can ...
2
votes
0answers
25 views

How to test a hypothesis which compares set of pairs of statements?

I've conducted an experiment but I'm not sure how to proceed with statistical analysis of it. I have pairs of sentences created by two groups of people A and B, semantically the sentences in each ...
2
votes
1answer
55 views

if $X$ and $Y$ are i.i.d., and if $X+Y$ and $X-Y$ are independent, are $X$ and $Y$ normally distributed?

Just recently come across Normal Distribution, and the following statement seems to be quite true, but is it? Can someone provide some general proof sketch if so please: For X and Y identically and ...
1
vote
1answer
42 views

Showing the normal distribution has points of inflections at $x = \mu \pm \sigma$ and a maximum at $x = \mu$

$X \sim N(\mu, \sigma^2)$ I.e. the density of $X$ is the normal distribution. I am looking to show that $f_X(x)$ has points of inflections at $x = \mu \pm \sigma$. In my notes it says that we ...
0
votes
1answer
51 views

Bringing a density in a normal distribution form

Because I do not want to exaggerate this thread Show that $E(Y\mid X=x)$ is a linear function in $x$ I continue my special problem here. In order to make the setting clear I'll give some information. ...