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

learn more… | top users | synonyms

0
votes
0answers
12 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\...
0
votes
1answer
27 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
votes
2answers
49 views

Why is the probability density function of a normal distribution exponential?

I came across this while self-studying for a probability course but I still don't quite get the rational behind it. Would appreciate it if someone could provide some intuitive explanation or rigorous ...
-2
votes
2answers
29 views

The derivative of the absolute value |x|

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
0answers
9 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 ...
0
votes
1answer
23 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 ...
1
vote
0answers
17 views

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

Probability with Martingales: approach 1: Assuming $$\lim E[\exp\{aS_n - bn\}] = E[\lim \exp\{aS_n - bn\}]$$ I can't seem to be able to prove $$\lim E[\exp\{aS_n - bn\}] = 0$$ with just $b &...
1
vote
0answers
45 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 ...
1
vote
0answers
34 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 ...
1
vote
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, \...
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 ...
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
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
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 ...
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} ...
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{...
1
vote
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 $f(x)$ ...
0
votes
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
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)}...
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-...
1
vote
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
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
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
32 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
votes
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
58 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 ...
0
votes
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
votes
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
votes
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 ...
0
votes
1answer
45 views

Determine values of $\mu$ and $\sigma^2$ for normally distributed random variable $X$

$X$ is a normally distributed (N($\mu$,$\sigma^2$)) random variable with $\mathbb{P}(X\leq0)=\frac{1}{3}$ and $\mathbb{P}(X\leq1)=\frac{2}{3}$. Determine values of $\mu$ and $\sigma^2$. I've tried ...
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 ...
2
votes
1answer
26 views

Probability - normal distribution

I apologize if this is too simple, but I just can't visualize how to get to the correct answer: I have a normal distribution of $X$ with $\sigma=5$, given that $P[X<35]=0.015$, find the mean of ...
3
votes
0answers
26 views

normal approximation of binomial distribution (“overbooking”)

for the following example of an "Overbooking" I have to calculate the probability by using the Central Limit Theorem: An airline books 52 seats whilst there are only 50 seats available. A guest ...
0
votes
0answers
20 views

Normal Distribution - Word questions

I am having trouble with the below question" The time spent waiting for a prescription to be prepared at a chemist's shop is normally distributed with mean 15 minutes and standard deviation 2.8 ...
0
votes
0answers
45 views

Algebraic structure of Gaussian PDFs

The answer to this question shows that the product of two Gaussian PDFs is also a Gaussian PDF. My questions are: Is there a multiplicative identity for this product? More generally what algebraic ...
0
votes
0answers
12 views

Did I implement a log-normal distribution incorrectly?

I have a bunch of data relating to the lengths of certain objects. I found out that the log of these lengths are somewhat normally distributed, so I used a lognormal distribution to approximate the ...
0
votes
1answer
32 views

Convergence in probability of a sum of dependent random variables to 0 [closed]

Suppose we know that $A_n + B_n \xrightarrow{p} 0$ as $n \rightarrow \infty$, and we know that $A_n \xrightarrow{d} N(0,1)$. Can we say that $B_n \xrightarrow{d} N(0,1)$? Note that $A_n$ and $B_n$ are ...
0
votes
1answer
72 views

The inequality related with normal distribution

When $Z \sim N(0,1)$, then $P(\vert Z\vert \gt t)\le\sqrt\frac{2}{\pi}\times \frac{exp(-t^2/2)}t$. My question is: I can get the aboving inequality, but how can I get the inequality such that $P(\...
0
votes
1answer
36 views

Hard time factoring Normal Distribution based on transformation problem.

My professor gave problems out to practice for our final on Wednesday. This problem is based on the transformation of two random variables. It a 5 part problem, so I will list the necessary portions ...
0
votes
1answer
35 views

Bivariate normal distributed vector $X (X,Y)$. Show distribution of $(X-Y, X+Y)$.

I have a a bivariate normally distributed random vector $X = (X,Y)$ and with Expected Value $(X)= (\mu(x),\mu(y))$, and Covariance Matrix $2\times 2$. (not independent) Now I want to show which ...
1
vote
1answer
21 views

Mathematical formula for equal distribution of amount among different group [closed]

Please let me know if you think i should edit my question or description. Problem statement: lets say i have spent $x on a sports material which needs to shared among total y memeber of the team. But ...
1
vote
1answer
33 views

Find exact difference between two values in Normal Distribution

If we have a normal distribution of N(10,2) and we are asked on what is the proportion of values betwen 7 and 8 we can calculate this by: ...
1
vote
0answers
45 views

Weak convergence of Poisson distributed random variables

I am stucked in the middle of an exercise: Let $$X_n,Y_m$$ independent random variables having the Poisson distribution with parameters n and m respectively. Show that $$\frac{(X_n-n)-(Y_m-m)}{\sqrt{...
1
vote
1answer
36 views

Compute expectation and covariance of Brownian bridge

Let $\{X(t), t \geqslant 0\}$ be a standard Brownian motion. That is, for every $t \gt 0$, $X(t)$ is normally distributed with mean $0$ and variance $t$. Then $\{X(t), 0 \leqslant t \leqslant 1 | X(1)...
3
votes
0answers
37 views

Why is $\dfrac{b(3x)}{b(x\bigoplus2x)}$ almost normally distributed?

I'm sorry if my question is a bit vague; I don't know a whole lot about distributions. Let $b(x)$ be the number of ones in the binary representation of $x$. I use $\bigoplus$ as bitwise XOR operator. ...
0
votes
1answer
34 views

On the sum of two independient normal random variables

Theorem. If $X$ and $Y$ are two independent normal random variables with means $a,b$ and variances $c,d$ respectly, the sum $X+Y$ is a normal random variable with parameters $a+b$ and $c+d$. My ...
0
votes
2answers
28 views

Chebyshev's inequality to find probability of interval

Here is how I solved the problem: $$ X\sim N(\mu=.13, \sigma^2=.005^2)\\ .12\le x\le .14 \\ \mu-2\sigma\le x \le \mu+2\sigma\\ $$ Using Tchebychev's inequality, I get $$ P(|x-\mu|\le 2\sigma)=1-\frac ...
0
votes
2answers
33 views

Are gaussian functions that have different kernel parameters orthogonal to each other?

If we have n gaussians where they have different scale and location parameters -- are they orthogonal to each other? By orthogonal I mean that the inner product is zero -- like it is for two cosine ...