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

learn more… | top users | synonyms

1
vote
1answer
34 views

What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
0
votes
1answer
24 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
1
vote
2answers
29 views

Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
1
vote
1answer
46 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
3
votes
1answer
53 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
0
votes
1answer
51 views

When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
2
votes
1answer
30 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
1
vote
2answers
45 views

How to show that normal random variables U1 and U2 are independent?

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
3
votes
1answer
44 views

Conditional probability for two normal distributed variables.

I haven't had to do much with probabilities since university, so please excuse if this is trivial or the question is not well specified. Let $X$ and $Y$ be two independent, normally distributed ...
1
vote
1answer
22 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
1
vote
1answer
22 views

Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... ...
0
votes
1answer
43 views

Word problem related to normal distribution

Consider the following problem: WINK, Inc. is made up of 450 employees who work a total of 13,500 hours per week. If the number of weekly work hours per person has a normal ...
0
votes
1answer
80 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
1
vote
1answer
27 views

Expected value of lognormal distribution.

Hi I'm stuck on this question: Recall that X is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if log(X) is normal with mean $\mu$ and variance $\sigma^2$. Suppose X is ...
0
votes
0answers
22 views

normal approximation to poisson

Cotton yarn is wound onto bobbins, each of which takes $100$m of yarn. If the thread breaks before $100$m is reached, the bobbin is rejected. In a trial of a new spinning machine, $13$ bobbins out of ...
-1
votes
2answers
45 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & ...
0
votes
0answers
27 views

Why we can use normal distribution to approximate binomial distribution when n is large enough?

Prove why we can use normal distribution to approximate binomial distribution when n is large enough. Hint: Try to read something on bernoull ...
0
votes
0answers
19 views

questions about distribution of multivariate normal

I'm looking at this past exam question, For A) Cbhat~N(CU,C(summation)C') B)I have very faint idea of what to do, I tried finding some theroems about distribution but couldn't find any that ...
2
votes
2answers
95 views

Evaluating Integral involving exp

I am stuck at the following integral :- $$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$$ Can anybody give me some hint. and also for this function $$ \int_{- ...
2
votes
1answer
16 views

Probability of an event occurring more than X% of the time

Cans are a normal random variable with a mean of 7.96 ounces and a standard deviation of 0.22 ounces. Suppose that you draw a random sample of 34 cans. Use normal approximation to find the ...
0
votes
1answer
29 views

Transforming a normal distribution to a uniform one

I'm searching for a method that transforms a normal distribution into a normal distribution. I've looked everywhere, but I'm not sure if I just missed something completely obvious, if this actually is ...
1
vote
0answers
18 views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
0
votes
2answers
33 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
0
votes
1answer
35 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
0
votes
1answer
27 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
1
vote
1answer
33 views

Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
0
votes
0answers
19 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
0
votes
1answer
16 views

Identity involving the relation Normal Distribution and Other arbritary Distribution

Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ ...
2
votes
2answers
31 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0$ + integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
0
votes
1answer
33 views

Expectation of random varible with normal distribution composed with exponential [duplicate]

I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} ...
0
votes
1answer
30 views

How to differentiate the standard normal deviation w.r.t. a parameter inside the upper bound

Given that $$N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{s^2}{2}}\:ds$$ And that $$d=\frac{1}{\sigma\sqrt{\tau}}\ln\left({\frac{S}{e^{-r\tau}K}}\right)+\sigma\sqrt{\tau}$$ How do I take the ...
0
votes
2answers
20 views

Is it correct that the normal approximation is just approximation of the normal distribution?

In mathematics statistics. I'm a bit confused by the terminology normal approximation. What is it? Is it just something you say when you approximate, for example the normal distribution?
0
votes
1answer
14 views

Normal distribution problem; distribution of height

The problem is: the height of children in age from 3.5 to 4 years is described by normal distribution with parameters $\mu =103$ centimetres and $ \sigma=4.5$ centimetres. What is the percent of ...
1
vote
0answers
24 views

how to prove $\mathop {\lim }\limits_{n \to \infty } {\{\Phi [(1 - \varepsilon )\sqrt {2\log n} ]\}^n}=0$?

$\Phi (x)$ is the distribution function of standard normal distribution. $\varepsilon$ is some positive tiny number that is less than 1. How to prove this beautiful and important limitation: ...
0
votes
1answer
20 views

Let $Y_{1},Y_{2},…,Y_{n}$ be a normal distribution where $\mu =2$ and $\sigma = 4$. Find $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$

Let $Y_{1},Y_{2},...,Y_{n}$ be a random sample from a normal distribution where the mean is $2$ and the variance is $4$. How large must $n$ be in order that $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$. ...
0
votes
1answer
28 views

problem regarding multiple random variables and normal distribution

I try to solve the following out of an old book on statistics: Cardboard boxes are stacked. The boxes have an average height of 10 cm and the height is normally distributed with a standard ...
1
vote
1answer
21 views

If the tubes are shipped in boxes of $1000$, how many wrong-sized tubes per box can doctors expect to find?

The cross-section area of plastic tubing for use in pulmonary resuscitators in normally distributed with $\mu = 12.5mm^{2} $ and $\sigma = 0.2 mm^{2}$ . When the area is less than $12 mm^{2}$ or ...
1
vote
0answers
19 views

Explanation of Approximation for Integral Over Gaussian Distribution

I am reading an optics textbook that uses the following integral to evaluate the power squared in the lower tail of the following Gaussian integral. $$\frac{1}{{{\sigma _P} \cdot \sqrt {2 \cdot \pi } ...
2
votes
1answer
40 views

In a game, $0.38$ buy hotdogs, how large an order should she place if she wants to have no more that a 20% chance of demand exceeding supply?.

A sell-out crowd of 42,200 is expected at Cleveland's Jacobs Field for next Tuesday's game with the Baltimore Orioles, the last before a long road trip. The ballpark's records from games played either ...
1
vote
1answer
35 views

Probability that out of their next 100 free throws, they will make between $75$ and $80$, inclusive in basketball game.

State Tech's basketball team, the Fighting Logarithms, have 70% foul-shooting percentage. (a) Write a formula for the exact probability that out of their next 100 free throws, they will make between ...
1
vote
1answer
27 views

68–95–99.7 rule mean normal distribution

if I have data that satisfy 68–95–99.7 rule, does it mean the data is normally distributed? Thanks
0
votes
0answers
23 views

General problem regarding normal distributions.

Let $X$ be the property of an item and let $X$~$N(\mu,\sigma=1)$. Now let $Y$ be the profit made from selling this item and let : $Y = \begin{cases} -R_1, & \text{if $X\lt6$} \\ C, & ...
1
vote
1answer
17 views

Standardization of a random variable with normal distribution

Let $$X \sim N(\mu, \sigma^2)$$ \begin{align*} P\bigg(\frac{X - \mu}{\sigma} \leqslant x\bigg) &= P(X \leqslant x\cdot\sigma + \mu)\\ &= \int_{-\infty}^{x\cdot\sigma + ...
1
vote
2answers
42 views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
1
vote
0answers
21 views

Empirical Rule. Is it applicable in this case?

So I ran in this problem: I have to test whether Empirical Rule is applicable. Proportions I got is 73%, 94,7% and 99.1% (within one, two and three standard deviations). I'm worried about 73%. This is ...
0
votes
0answers
20 views

Relative Error of Normal Approximation

I have this math statistics assignment that I worked out and came out with a couple of graphs. How do I interpret the graphs? Task: If you consider the average of 20 iid rvs, normal distribution is ...
0
votes
1answer
33 views

Student's distibution

If $X_i$ are independent equally distributed random variables, $S_n=X_1+...+X_n$, then $$\frac{S_n-n\mathbb E(X)}{\sqrt{n \sigma^2(X)}}$$ tends by distribution to $N(0,1)$ for $n \to \infty$. It is ...
1
vote
0answers
26 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
3
votes
0answers
23 views

$L^2$ limit of Gaussian random variables

Let $X_n$ be a sequence of Gaussian random variables defined on the same probability space. The statement is that if $X_n$ converges to some random variable $X$ in $L^2$-sense, then $X$ is also ...
0
votes
1answer
35 views

Exponential distributed with expected value

Have this question in math statistics Normal Distribution. In a certain cellular phone system new calls arrives with exponential distributed interarrivaltimes with expectation value $$\mu ...