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

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4
votes
1answer
97 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 ...
0
votes
1answer
34 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] = ...
0
votes
1answer
52 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. ...
0
votes
1answer
27 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, ...
1
vote
1answer
110 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 ...
1
vote
2answers
38 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, ...
0
votes
0answers
24 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 ...
0
votes
1answer
43 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 ...
0
votes
1answer
38 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 ...
6
votes
4answers
104 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 $E(X^2)=1$?
0
votes
2answers
64 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 ...
1
vote
1answer
29 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 ...
1
vote
1answer
44 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. ...
2
votes
1answer
43 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 ...
1
vote
1answer
57 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 ...
0
votes
0answers
34 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 $F(x)$. $$\int_{T}^{\infty} xf(x)dx $$ ...
0
votes
1answer
60 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 ...
1
vote
1answer
201 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 ...
1
vote
2answers
151 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 ...
0
votes
0answers
33 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
1
vote
2answers
67 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. ...
3
votes
0answers
171 views

PDF of X +Y + X* Y, when X and Y are independent Normal [closed]

I have $X,Y$ iid Normals $N(0,\sigma^2)$ What is the distribution of $X+Y+YX$? Thnks a lot!
1
vote
1answer
41 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 ...
0
votes
2answers
61 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 $10$ ...
3
votes
0answers
27 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 $F_{Y_n}$ ...
0
votes
1answer
80 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 ...
4
votes
2answers
109 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! ...
1
vote
2answers
35 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 ...
1
vote
1answer
42 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 ...
0
votes
1answer
45 views

stdev and mean from gaussian fit vs. from classical formula

I have a set of data - measured speed of molecules in water. I made a histogram and fitted it with function $$A\exp\frac{(x-B)^2}{C}$$ calculating mean and standard deviation from values B and C If I ...
1
vote
0answers
25 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
0
votes
0answers
17 views

Pivotal quantity that is a function of the z-score: find the CI

** Assumptions ** Let: $X$ be a random variable. $\bar{X}_n$ be the sample mean of X; $\mu$ be the expectation value of X; Assume that $\mu$ is not observable; $S_n^2$ be the sample variance of ...
0
votes
1answer
97 views

Statistics: Relationship between process capability and mean

A company produces one-kilogram sugar packets. The specifications on the net content are 1000 ≠ 5 grams. Assuming that the net content follows normal distribution with mean weight as 1005 grams and ...
0
votes
0answers
19 views

When $(X, Y)$ are jointly normal given that both $X$ and $Y$ are normal?

We know that $X$, $Y$ are normal does not guarantee $(X, Y)$ is jointly normal. A typical example is: $X=Z$, and $Y=ZU$, where $Z$ standard normal, $P(U=1)=P(U=-1)=1/2$, and $Z, U$ are independent. ...
1
vote
0answers
48 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
3
votes
1answer
135 views

Finding the distribution function of a random variable using CLT

Let $f_0$ and $f_1$ be two continuous probability density functions with means $\mu_0,\mu_1$ and variances $\sigma_0^2,\sigma_1^2$ on $\mathbb{R}$. Furthermore, let $l(y)=f_1(y)/f_0(y)$ be the ...
1
vote
1answer
82 views

Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
0
votes
1answer
13 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...
0
votes
0answers
7 views

Finding a Gaussian Distribution to approximate a distribution with non-positive definite covariance matrix

We have got a Gaussian distribution covariance matrix(precision matrix) and the potential information, that is, if g is proportional to exp(-X'KX+h'X). However, K here is not positive definite. So we ...
0
votes
1answer
21 views

Moments of maximum of bivariate standard normal

Let $X,Y \sim N(0,0,1,1,\rho): f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}}$, and let $Z=max\{X,Y\}$. I'm looking for the first two moments of $Z$. I know it is ...
0
votes
2answers
107 views

normal distribution using Z - finding probability between 2 numbers

I am wanting to find the probability of the following: SD = 20 Mean = 100 P(85 < X < 117) i have found the z values for both: P(X>85) : X-u/o = 85-100/20 Z = -0.75 and found the ...
2
votes
1answer
47 views
0
votes
2answers
30 views

Understanding sampling from a normal distribution with zero mean

I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the ...
0
votes
1answer
29 views

Bivariate normal distribution when $\rho$ is 0

What happens to the bivariate normal distribution when $\rho$ is 0?The bi-variate normal reduces to a simpler distribution, but what is it? and how do you calculate the cdf then? What I have tried: ...
1
vote
1answer
58 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
0
votes
3answers
108 views

Gaussian integral evaluation

Asked a question to evaluate the Gaussian Integral, $$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$ using the the following approximation, $J=\Bbb E[X^2] \sim J_N = 1/N ...
0
votes
2answers
52 views

Normal Distribution Problem

The time taken for a computer to connect to a server is normally distributed with a mean value given by 3.3 seconds and a standard deviation of 0.66 seconds. (a) A computer is said to have a fast ...
1
vote
0answers
35 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
0
votes
1answer
55 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$