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

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1answer
19 views

Normal Distribution how $N(x-x_n|0,\sigma^2) = N(x |x_n,\sigma^2) $

I read an expression Could someone explain the step $N(t-t_n|0,\sigma^2) = N(t | t_n,\sigma^2) $ ?
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1answer
73 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 ...
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0answers
22 views

What is the minimum standard deviation for a normal PDF such that one tail is always larger than that of a second normal PDF (different means)?

Say I have two weighted normal distributions, $$ f_1(x) = \frac{a}{2 \sigma_1} e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}} $$ and $$ f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} $$ ...
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1answer
96 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 ...
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0answers
130 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
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1answer
39 views

Distribution of distance from 0 of gaussian point

Suppose $X_1,...,X_d\sim\mathcal{N}(0,1)$ are i.i.d.'s, each distributed normally around 0 with variation 1. It looks like $\mathbb{E}\left(\sum X_i^2\right)=d$. Why is that true? And how $Y=\sum ...
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1answer
246 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 ($\operatorname{E}[Y] = 0$) ...
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1answer
87 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, ...
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1answer
25 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 ...
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2answers
136 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
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1answer
319 views

Central Limit Theorem for uncorrelated (non-independent) but bounded random variables

Given uncorrelated, discrete random variables $X_i$ that are bounded, e.g., they can only take on values $|X_i| \leq 4$, then is there a form of the central limit theorem that one can apply to the ...
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1answer
53 views

How to find $E[X|X>Y]$

Suppose $X$ and $Y$ are independent standard normal variable. I want to find $E[X|X>Y]$. I calculated that $$ f_{X>Y}(x) = 2\Phi(x)\phi(x)$$ However I couldn't find the expected value using ...
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1answer
26 views

Standard normal RV probability

Z is a standard normal random variable I need to find $P(|Z|<.95)$, find c such that $P(|Z|<c)$, and given that X is a RV with mean 3 and standard deviation 16, find $P(X>3.84)$ I am just ...
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0answers
21 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 ...
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1answer
48 views

Distribution of log-log linear regression

Edit: Sorry yeah not too clear, probably posted this too late at night... Essentially I have data which appears to be in exponential form - a log-log graph put it close to a straight line. Using R, I ...
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1answer
176 views

Is normalcdf() inclusive?

I was looking at these examples here: Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is ...
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0answers
55 views

How to simplify the computation of a special case of multivariate normal cdf

I am trying to compute a multivariate normal cdf where all but the last bounds of the integrals are symmetric: $$F(a, \sigma, m ) = ...
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0answers
21 views

Continuity Correction with replacement

An urn contains 2 white and 8 red marbles. A marble is drawn from the urn 100 times in succession with replacement. What is the probability of drawing more than 75 red marbles? My attempt: $n=100, ...
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0answers
38 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 ...
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1answer
49 views

If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note ...
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0answers
40 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 ...
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1answer
93 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 ...
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1answer
95 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 ...
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1answer
110 views

Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
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1answer
76 views

upper and lower bounds of $E[X|X>x]$

I am trying to find tight upper and lower bounds for $E[X | X > x]$ where $X$ follows a standard normal random variable. After calculations I found that $$ E [X|X>x] = ...
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1answer
54 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. ...
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1answer
31 views

Variance algebra

This might seem very simple but I'm having some trouble getting to the answer. If I have a random variable that's normally distributed $$X\sim N(30, 3^2)$$ and another random var. $$Y \sim N(20, ...
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1answer
512 views

Intuition and the math behind normalization

What exactly is the purpose of normalization. From what I read, it is to adjust two different sets of values so you can compare them, but I don't understand why, nor the math behind it. Could anyone ...
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2answers
49 views

Using continuity correction for normal distribution

Suppose a fair coin is tossed $900$ times. Find the probability of getting more than $475$ heads. Use the continuity correction. My answer: $n=900, p=1/2, q=1/2$ $\mu=900(1/2)=450, ...
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35 views

Average minimum distance

Let $\mathbf{u} =\begin{bmatrix}u_1 & u_2 & \dots & u_N \end{bmatrix}^T$ and $\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \dots & v_N\end{bmatrix}^T$. All the elements of ...
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1answer
72 views

probability, normal distribution mean [closed]

Should I use a certain table for this question or should I use a special formula. A random value has a normal distribution with the mean 102.9 and the standard deviation 4.7. What are the ...
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1answer
63 views

Chi square distribution vs. Chi square test

I am trying to link my understanding of the Chi square test with my conception of the chi square distribution. More precisely - i understand the procedure of the chi square test, e.g. as when used ...
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4answers
109 views

If $X \sim N(0,1)$, why is $E(X^2)=1$?

If $X$ is a normally distributed with mean $0$ and variance $1$, expectation of $X$ equals $0$ but why is $E(X^2)=1$?
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2answers
85 views

Painful? Moment Generating Function

Part 1 Let $X$ be a random variable with the p.d.f. $f(x)=\frac{1}{4\pi}e^{\frac{-x^2}{4}}$, compute the MGF of $X$. So I know I want ...
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1answer
44 views

Integral of cumulative normal

Let $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$ Question: for what values of $a$, $b$ and for what choices of $f(x)$ would the following ...
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1answer
67 views

Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
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1answer
61 views

Source needed: Does asymptotic normality yield asymptotic unbiasedness and consistency?

Assume that $$\sqrt{n}(\hat g - g(\theta)) \xrightarrow{d} Z, $$ where $Z$ is $N(0,\sigma^2)$. Does this already imply asymptotic unbiasedness and/or consistency, i.e., $$ E[\hat g] \rightarrow ...
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1answer
81 views

Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
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0answers
88 views

Integral with truncated normal distribution

I am attempting to determine closed form equations for several integrals. Suppose $X=N(\mu,\sigma)$ is normally distributed with PDF $f(x)$ and CDF $F(x)$. $$\int_{T}^{\infty} xf(x)dx $$ ...
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1answer
77 views

Normal Ratio Distribution with CDF Method

I think I'm missing something glaringly obvious here that's causing problems for me in the entire subject. I have two independent standard normal random variables, X and Y ~N(0,1), and I need to find ...
2
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1answer
768 views

Distribution of the sum of normal random variables

Let $X\sim \mathcal N(\mu_X,\sigma_X^2),\ Y\sim \mathcal N(\mu_Y,\sigma_Y^2)$ two normal random variables and $a,b\in \mathbb R$. If $X,Y$ are independent, then $$aX+bY\sim \mathcal ...
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3answers
1k views

Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
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2answers
70 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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1answer
49 views

The probability that a joint distribution is less than a certain value, given the correlation coefficient.

For this problem, we are told that $X$ and $Y$ are jointly normally distributed variables, both being standard normal. We're given their correlation coefficient. So, how do I get from there to finding ...
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2answers
242 views

Normal distribution squared probability

Let $X_1,X_2,X_3,X_4$ be independent standard normal random variables and $Y=X^2_1+X^2_2+X^2_3+X^2_4$. Find the probability that $Y≤3$. Enter your answer as a decimal and make sure that at least $10$ ...
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0answers
32 views

Write $\Phi_n(\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. ($\Phi_n$ CDF of a $\mathcal{N}(0,\frac{1}{n})$)

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$ Let $F_{Y_n}$ ...
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1answer
109 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
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2answers
117 views

Fraction Problem. 3rd grader question got parents thinking

So our nine year old son comes home from 3rd grade and tells us an amazing thing happened in school today. He was playing a math game with his friend and they got the same score two times in a row! ...
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2answers
46 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
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1answer
53 views

Mean and variance: Gaussian is the most conservative assumption

"given only the mean and variance of a distribution, the most conservative assumption that can be made about the distribution is that it is a Gaussian having the given mean and variance" I've read ...