0
votes
1answer
22 views

If X ~ N(0,σ^2), find the density of Y = |x|

If X ~ N(0,σ^2), find the density of Y = |x| Hi I am reviewing for an upcoming exam, and came across this question in the textbook. Can someone please help me with this question. Thanks
0
votes
1answer
22 views

Intuition behind Normal distribution forumula

In this formula $$ P(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{ - \frac{ \left( {x - \mu } \right)^2 }{2\sigma^2}} $$ why do we divide by square root 2 pi and after that multiply everything by e in ...
0
votes
1answer
11 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
-1
votes
0answers
19 views

IQR of the sampling distribution of S^2

Suppose $X_1,X_2,...,X_5$ is a random sample from a $N(33,9)$ population and let $S^2$ be the sample variance. [a] Find the IQR of the sampling distribution of $S^2$. [b] Find the IQR of the ...
0
votes
1answer
27 views

Standard Normal Distribution and CDF

I have a data set which consists of measured time in seconds. Secs= ${3000, 3857, 2400, 3323}.$ Mean $\mu =3145$. Standard deviation $\sigma=609.556$. I calculated the Standard Normal variable for ...
1
vote
0answers
19 views

Log-likelihood of the normal distribution.

On the attached picture I've highlighted the term which I do not agree with. Is it actually true ? In my calculations I get $$-n(\frac{1}{2}\log(\sqrt{2\pi})+\log(\sigma)),$$ instead. Thank you in ...
0
votes
1answer
32 views

Beginner Econometrics question about probabilities for a normal variable

$Y \sim N(\mu, \sigma^2)\implies (Y-\mu)/\sigma$ Prove that this has a Mean of $0$ and a Variance of $1$.
1
vote
1answer
13 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
0
votes
2answers
44 views

Proof that if $Z$ is standard normal, then Z^2 is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of ...
1
vote
1answer
44 views

Finding the 99% of a normally distributed graph

The heights of adults are normally distributed with a mean of 187.5 cm and a standard deviation of 9.5 cm. A standard doorway is designed so that 99% of adults have a space of at least 17 cm over ...
1
vote
4answers
33 views

when to use which z score equation?

in some exam past papers I have been doing I have come across the statistics equation z=(sample mean - mean)/standard deviation as well as the equation z = (sample mean - mean)/(standard ...
1
vote
2answers
31 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
0
votes
1answer
20 views

Standard deviation in normal distribution

A manufacturer uses a machine to make metal rods.The diameter of the rods follow a normal distribution with a mean of 1cm and a standard deviation of 0.02cm If the standard deviation of the diameters ...
1
vote
0answers
276 views

Outliers in a Normal Distribution

Im doing AP Stat. in High School level. Here is a question i am stumped on because i feel like it is maybe a threory or law or something that i just never learned. However it DOES ask to show my ...
0
votes
1answer
18 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
0
votes
2answers
28 views

Normal distribution and algebra problem

Bags of cement are labeled $25 \operatorname{kg}$. The bags are filled by machine and the actual weights are normally distributed with mean 26.0 kg and standard deviation $0.50 \operatorname{kg}$. It ...
1
vote
2answers
42 views

Why we consider log likelihood instead of Likelihood in Gaussian Distribution

I am reading Gaussian Distribution from a machine learning book. It states that - We shall determine values for the unknown parameters mu and sigma^2 in the gaussian by maximizing the ...
0
votes
0answers
34 views

Determining $\sigma$ given mean and proportion of a Normal distribution?

The marks of a random sample of students with mean $\mu$ and standard deviation $\sigma$ showed that 15.87% scored higher than 70. The distribution of the marks is Normal with mean $50$ standard ...
1
vote
1answer
16 views

Solving a statistics equation

Suppose $X$ is a random variable which follows a Poisson distribution, such that, for some positive integer $m$, $$X \sim Po(0.01m)$$ Find the least value of $m$ such that $$P(X \ge 1) > 0.9$$ ...
0
votes
1answer
11 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
63 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
26 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
24 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
71 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
2answers
29 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?
1
vote
1answer
40 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
91 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
37 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
53 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
21 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
13 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
17 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
55 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
35 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
67 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 ...
-5
votes
1answer
40 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
71 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
38 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
37 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
103 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
40 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
24 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
20 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
74 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
141 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
59 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
31 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
100 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
14 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 $\sigma$ ...