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

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0
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2answers
22 views

Normal distribution and choosing final number

I've read a paper which used normal distribution in order to assign a number to each entity as follows: Each user has a quality measurement qi ∈ [0, 1]. For the experiments in this paper, the ...
1
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2answers
21 views

How to calculate inverse cumulative distribution using a table?

I need help with this: $$P(X\geq a)=1-F_X(a)=1-\Phi\left(\frac{a-70}{8}\right)=0.25$$ When $X\sim N(70,64)$. I know that it should be: $(a-70)/8 = 0.6745$ How do I get $0.6745$ From $Z$ table? I ...
0
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0answers
17 views

Significance and Poisson processes

Must I include the Poisson error on a new observation when considering whether it is consistent with a distribution? For example, the number of rain drops that fall into a cup per minute are recorded....
1
vote
1answer
45 views

Generalized law of large numbers

Do you know any kind of generalisation of the law of large numbers. I mean something like this : Assume that $(X_n)_{n\in\mathbb{N}}$ is a sequence of independant variables (not necessarily ...
0
votes
1answer
31 views

Show how Gaussian with mean $\mu$ and std dev $\sigma$ is constructed.

I am a computer science student, who has recently taken an interest in data science. I've been learning about Gaussian distributions, and I've read from the documentation of the numpy python package ...
1
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2answers
59 views

Why does the normal distribution describe data collected in real life so well? [on hold]

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$ Is there any intuition behind choosing $e^{-x^2}$ instead of some other function?
1
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1answer
36 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
2
votes
0answers
31 views

Conditional expectation of independent normals

As it was proved in the answer here, for $Z_1, Z_2$, two independent and identically distributed random variables. Then we have: $$ \mathbb E[Z_1\mid Z_1+Z_2] =\frac{Z_1+Z_2}{2}. $$ However, suppose ...
0
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0answers
12 views

Existance of a UMVUE [on hold]

$ \{X_{i}: 1\leq i \leq n \} $ is a random sample, i.i.d $ N(\mu, 1) $ with $ \mu $ unknown. For a fixed $ x_{0} $, does there exist a UMVUE for $ \phi(x_{0}-\mu) $, where $ \phi $ denotes standard ...
0
votes
3answers
40 views

Normal distribution

can anyone help me calculate $E(Z^4)$, $E(Z^3)$ for $Z\sim N(0,1)$? I know that $Z^2\sim \chi^2(1)$ then $E(Z^2)=1$, $Var(Z^2)=2$. Thank you.
0
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0answers
13 views

The finite-dimensional distribution of a stochastic process

Let $K(s,t)$ be a real function over $T\times T$, where $T$ is arbitrary. $K$ has two properties: $K$ is symmetric ($K(s,t)=K(t,s)$). $K$ is nonnegative-definite ($\sum_{i,j=1}^k K(t_i,t_j)x_ix_j\...
-3
votes
2answers
36 views

The derivative of the absolute value |x| [duplicate]

I read about the derivative of the absolute value |x|, but why the absolute value is not differentiable at point zero, and when it becomes 1 or -1 {geometrically}? Thanks
0
votes
1answer
28 views

Conditions for Normal Approximation to Binomial

It is well known that if $np > 5$ and $n(1-p) > 5$ that a normal distribution with mean $np$ and variance $np(1-p)$ can be used to approximate a binomial distribution. My question is, what ...
0
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0answers
11 views

Conditional Normal Distribution given Another Normal Distribution

Say I am interested in the following condition distribution $Y|X=x \sim N(1-aX, bX^2)$ and $X = f(x)$ where $x \sim N(0,1)$ How would I be able to determine the distribution of $Y$. I am assuming ...
5
votes
5answers
19k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
1
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0answers
51 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
0
votes
1answer
54 views

Deriving MLE for covariance matrix using Robbins-Monro

I'm having some trouble completing exercise 2.37 in Bishop's Pattern Recognition and Machine Learning text. I'm not reading this text as part of a course, so this is not a homework question. Here's a ...
3
votes
2answers
2k 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
35 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
1
vote
0answers
15 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
0
votes
1answer
21 views

Gaussian expected inner product with respect to fixed matrix

Suppose $R\in\mathbb{R}^{p\times p}$ is a a fixed matrix (it can be asymmetric, non-positive definite, and so on). Then, I would like to find a formula for the expectation $$\mathbb{E}_{x\sim N_p(0,V)...
1
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0answers
22 views

Quotient of two Gaussian densities

The matrix cookbook contains formulas for the product of two multivariate Gaussians, but doesn't appear to contain formulas for the quotient of two Gaussians. $$ \frac{\mathcal{N}(\mathbf{m}_1, \...
2
votes
2answers
41 views

Sum and covariance of standard normal distribution

If $X \sim \mathcal N(0,1)$ and $Y \sim \mathcal N(0,1)$ are i.i.d. standard normal distributed how can I find: $W=3X+Y-2$ $\mathrm{Cov}(X+Y, X-Y)$ $\mathbb{P}(X\lt2Y)$ Q1 Not sure: $W\sim\...
0
votes
1answer
28 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [closed]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
0
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0answers
19 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
0
votes
3answers
68 views

Compute $\mathbb{P}(1<X^2+Y^2<2)$ when $(X,Y)$ is i.i.d. standard normal

Assume that $(X,Y)$ is i.i.d. standard normal. Compute $\mathbb{P}(1<X^2+Y^2<2)$. So I've decided to use polar coordinates to solve and I've gotten to this point: $$\iint_{1\lt X^2+Y^2\lt2} ...
0
votes
1answer
514 views

Gaussian distribution and its parameters

I need to learn more about Gaussian distribution and given a set of data, plot a Gaussian distribution of it. Using the following code sample, could you please tell me how I can plot a Gaussian ...
0
votes
1answer
19 views

error term in time-series Seasonal AR model

I am reading a paper related to timeseries forecasting in which I have a question regarding the seasonal AR model described in equation (1.2) namely: $log(y_t)$~$log(y_{t-1}) + log(y_{t-12}) + x^{(1)}...
2
votes
1answer
76 views

Compute the probability of a joint event involving two independent standard normals

Suppose $X$ and $Y$ are independent, standard normal random variables. I'm trying to compute the probability of the event $$ \{X \leq x, Y \leq kX\} $$ where $k$ is a positive constant. The ...
3
votes
1answer
73 views

Help required in finding solution to overdetermined system of equations?

I have access to M probability measures, $P_e(c_1),P_e(c_2),\cdots,P_e(c_M)$, defined as \begin{equation} P_e(x) = p(x|y) = p(y|x)\cdot \mathbb{P}(X=x) \frac{1}{\sqrt{2\pi\sigma^2}} \exp\Big[-\frac{(y-...
0
votes
1answer
34 views

Probablity of normal distribution when x is a function

Assume a uniform distribution random variable X~U(0,1). And $\Phi$ is the symbol of the standard normal distribution. Assume $Y=\Phi^{-1}(X)$. The question is, $\mathbb{P}(Y \le 0)=?$. The Solution is ...
1
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0answers
35 views

Existence of solution of equation involving normal distribution

I've tried to show that the following equation has a solution: \begin{equation*} g(x)=\left[1-\left(2\int _{\mu}^{x}f(y)dy\right)^2\right]-8xf(x)\int _{\mu}^{x}f(y)dy=0, \end{equation*} where $f(x)$ ...
2
votes
1answer
466 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
3
votes
0answers
47 views

How to approximate the cumulative distribution function of the normal by a product of functions?

Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{...
0
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0answers
38 views

Central Limit Theorem for gambling return ratio

Consider a single bet with odds $o$ and thereby implied probability $1/o$. Assume that the real probability $p$ is known. Let $I$ be the stake, and $y$ the return from the bet. Then, $\mathbb{E}(y) ...
0
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1answer
24 views

Biostatistics Sampling Distributions

I am so confused, can someone please help? The activity of a certain enzyme is measured by counting emissions from a radioactively labeled molecule. For a given tissue specimen, the counts in ...
0
votes
1answer
510 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
1
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1answer
45 views

The standard deviation is more stable than the mean?

In an introduction to the subject of hypothesis testing, a book on probability and statistics for engineering students has a statement asserting that "the standard deviation is more stable than the ...
0
votes
1answer
17 views

Does standardizing a random variable that is not normally distributed change the underlying distribution?

For my analysis I am standardizing Response Times, which are usually known to be skewed and are in my data set, using the "classic" standardization method of substracting the grand mean and dividing ...
1
vote
2answers
27 views

Probability Sum of components > value

I have a question that I cannot find the method in finding the solution. Question: A device is made up of 5 subcomponents, denoted i=1,2,3,4,5. A subcomponents mean weight is 10i grams. All are ...
0
votes
1answer
35 views

Normal Distrubution

I am studying Normal Distrubution and stuck at one problem.pls help quest-If the mean of a normal frequency distribution of 1000 items is 25 and its standard deviation is 2.5, then its maximum ...
0
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0answers
16 views

Conditional Mean Given Precision Matrix While Avoiding Inversions

I'm working on a problem where I need to compute a conditional mean directly from a precision matrix (the inverse of covariance matrix). Let $\boldsymbol \mu$ be a mean vector partitioned into $$\...
2
votes
2answers
60 views

Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$

For standard normal distribution, the expected value of $x^2$ is $1$. A natural question is that in the multivariate case, what is the expected value of $x^t\Sigma x$ for multivariate normal ...
1
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0answers
64 views

If $X\mid Y$ and $Y$ are both normal, is $X\mid Y>y$ normal as well?

Consider two random variables, $X$ and $Y$, with the following properties: $X\mid Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X\mid Y>y$ follow a normal distribution as well? If so, what ...
1
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1answer
2k views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
0
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0answers
13 views

Where did I make a mistake in this transformation of random variable?

The arctangent of a standard Cauchy random variable $Z\sim\text{Cauchy}(0,1)$ is uniformly distributed in $[-\frac{\pi}{2},\frac{\pi}{2}]$. The proof is straightforward: $$P(\arctan(Z)\leq t)=P(Z\...
0
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0answers
20 views

Cumulative bivariate normal

How do I calculate the cumulative probability distribution function for a bivariate normal distribution with conditions $P( x>a , y>b)$? Is there any method to solve $$P(x>a,y>b)\\\int_{...
0
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0answers
8 views

A question about Estimation problem in digital communication setup.

I originally asked this problem here http://dsp.stackexchange.com/questions/31503/estimation-problem-for-m-ary-pam-transmission-over-awgn-channel-problem I would appreciate if someone can take a ...
1
vote
1answer
38 views

Normal Approximation - how many bookings so probability for “overbooking” stays under certain value

I need some help with the following: A hotel has $r$ rooms. The probability that a guest who booked a room also appears (which means: no cancellation) is $p = 0.9$. I'd like to know how many rooms ...
1
vote
1answer
579 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...