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

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1answer
21 views

How to eliminate coefficients from a sum

For given random values $$X_i \sim\mathcal{N}(0,1)$$ and $$\frac{X_i-\mu}{\sigma}=\tilde{X_i}\sim\mathcal{N}(\mu,\sigma),\,\mu\in\mathbb{R},\,\sigma>0$$ prove ...
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3answers
82 views

Solving integral related to second moment of normal distribution restricted to interval. CAS answer acceptable

Let $f(x)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$, the pdf of the 1-dimensional normal distribution. Is it possible to compute $\int_{-a}^a x^2 ...
0
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1answer
58 views

How to prove Gaussian integral in normal distribution can be scaled to a standard curve?

If I want to solve the gaussian integral for normal distribution problems I only need to scale it to a standard normal distribution curve and consult a table. I want to know why this is valid (the ...
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1answer
16 views

Conversion to standard normal

How can I convert a the pdf of a normal distribution that it N(t,1), but integrated from 0 to infinity, to the standard normal. I found that the former is equal to 1- ϕ(-t) but i cant figure how this ...
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1answer
33 views

Is the joint distribution of two independent, normally distributed random variables also normal?

Say I have $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, also $X$ and $Y$ are independent, then is the joint distribution of $X$ and $Y$ multivariate normal? I.e., $$\begin{bmatrix} ...
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1answer
41 views

Show that two sums have the same distribution?

I have not been able to show that the following two stochastic variables have the same distribution. My question is as follows: Let $$ X_1, X_2,..., X_n $$ be independent and identically ...
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1answer
24 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
0
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1answer
23 views

Calculating conditional mean of 2 Normal

If $\theta$ is $N(\bar{\theta},\sigma^2_\theta)$, and $s=\theta+\epsilon$, where $\epsilon$ is $N(0, \sigma^2_\epsilon)$, how can I derive that ...
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0answers
18 views

convergence of sequence of functions with finite second moment

Given $0<a<1$. Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ ...
2
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3answers
43 views

Why do we use only 1/2 for continuity correction in case of approximating binomial random varable to a standard normal random variable?

I have read about continuity correction in case of approximating a binomial random variable to a standard normal variable. But in all the examples , they only use 1/2 as a continuity correction ...
2
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0answers
33 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
3
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1answer
27 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
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0answers
37 views

Concentration of the absolute norm of a normal distribution given via this integral?

Let $\mu\sim N(0,\Sigma)$ be a multivariate Guassian distribution, with $\hat\mu$ the one-dimensional marginal of $\mu$. I'm not sure exactly what the one-dimensional marginal of $\mu$ is, but I think ...
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2answers
18 views

Does it matter here that random variables are jointly normally distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
0
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1answer
17 views

covariance matrix in bivariate distribution

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I ...
0
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1answer
61 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
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1answer
44 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
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0answers
26 views

Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
0
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1answer
69 views

Dirac Delta function and normal distribution

I understand the Dirac Delta is the limit of a normal distribution when the variance of the normal distribution tends to 0: $$ \delta(x) = \lim_{v\to 0}\frac{e^{-x^2/2v}}{\sqrt{2\pi v}} $$ Then what ...
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2answers
36 views

What is the expected value of a standard normal random variable given value is positive?

Am not sure if I'm wording this correctly. But say we take huge sample of standard normal random variables. Then we separate out positive values. What would be average of the positive values ? What ...
0
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1answer
12 views

Standardizing Normal Distribution

I was listening to Statistics lecture on Normal distribution and I could not understand that how P(X-mean)/S.D<=(x-mean/S.D) becomes \phi (x- mean/ SD) got solved by chain rule.
2
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1answer
86 views

Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population.

Recall that for a $N(\mu,\sigma^2)$ population $W=\frac{n-1}{\sigma^2}S^2\sim \chi^2(n-1)$. [a] Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population. ...
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0answers
8 views

How does one scale a covariance matrix learned on de-meaned and scaled data?

I have a dataset on which I want to train a multivariate mixture of gaussians. One common thing to do is de-mean and scale the data such that each feature has zero mean and unit covariance. If I ...
3
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1answer
87 views

Estimate large covariance matrix using few samples.

Let $\mathbf{x}$ be a random vector in $\Bbb{R}^n$, such that $\mathbf{x}\sim N(\bar{\mathbf{x}}, \Sigma)$. $N$ observations of $\mathbf{x}$ are available, say $\{\mathbf{x}_i, i=1,\ldots,N\}$. The ...
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1answer
18 views

Joint distribution of two gaussians, one of which is dependent on the other.

Suppose $x\sim N(\mu_x,\sigma_x^2)$ and we are given that $y\mid x \sim N(a+bx,\sigma^2)$, where $a$ and $b$ are some constants. It is a fact that the joint distribution of $x$ and $y$ is a bivariate ...
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1answer
28 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
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0answers
37 views

10 random observations

I have to generate $10$ random numbers with mean $= 22.6$ and standard deviation $= 2.4$. Then I have to see if the $10$ observations appear to be normally distributed. When I generate $10$ random ...
1
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1answer
39 views

Updating in game with normal distribution

In a game from the following paper, it is stated that Player $i$ observes a private signal $x_i = \theta + \epsilon_i$. Each $\epsilon_i$ is independently normally distributed with mean $0$ and ...
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0answers
22 views

Range of sum of Normal Distribution.

May be its silly question but I was just wondering is there any way to find out the absolute range of sum of values of Random normal distribution of N numbers with mu and sigma as mean and Std. Dev. ? ...
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0answers
19 views

Certain parasite Trypanosome

Consider the length of individual trypanosome chosen at random from the population. Find a) $\def\Pr{\operatorname{Pr}}\Pr\{20 < length < 30\}$ I just say 0.41 + 0.21 = 0.62 b) $\Pr\{length ...
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1answer
29 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
0
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1answer
50 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
1
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1answer
15 views

Normal distribution of juice

Quantity of juice in a pack of 1L is normaly distributed with average (mean) 950ml, and with standard deviation of 10ml. What is the probability that random pack of juice contains less then 945ml of ...
0
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1answer
46 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 ...
3
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1answer
55 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
0
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1answer
18 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 ...
0
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1answer
98 views

Do we need to use continuity correction if we use CLT to do normal approximation

In a hotel, large number of cups and saucers are washed everyday. The number of cups that are broken each day while washing averages $2.1$. The number of saucers broken each day averages $1.6$, ...
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0answers
11 views

Normal distribution around and extreme value

I'm trying to create an artificial dataset with users, items, and ratings given by the users to the items. Creating the dataset, I pick the average rating for every item randomly, and let the ratings ...
0
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1answer
36 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 ...
5
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1answer
232 views

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
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0answers
20 views

What type of distribution can be used to describe a game with positive expected winnings?

I've come across something I'm not too sure about. Let's say we flip a coin, heads mean we lose 1 unit, tails means we win a 1 unit. This distribution of outcomes in this would be considered normal, ...
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0answers
12 views

finding sampling distribution from standard normal

Let $X_1$, $X_2$, $X_3$ be a random sample of size $3$ from a standard normal distribution. Find the distribution of $X_1^2 + X_2^2 + X_3^2$. If possible, find the sampling distribution of ...
3
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1answer
62 views

Boundedness of an integral of square function implying zero integral

Let $\alpha:\mathbb R\mapsto\mathbb R$ be the smooth function such that $$\int_{-\infty}^{\infty}[\alpha'(x)-x\alpha(x)]^2e^{-\frac{x^2}2}dx<\infty.$$ I wish to prove that ...
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0answers
22 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 ...
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1answer
37 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 $1$.
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1answer
15 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
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2answers
76 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 ...
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0answers
51 views

Poisson process. Finding 5th and 95th centiles

I am an undergraduate student of Economics. Today I was trying to solve 1 exercise related to Poisson process that I found confusing and I would be very grateful for your help, as my Mathematics ...
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1answer
73 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
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1answer
51 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 ...