Tagged Questions
0
votes
0answers
26 views
Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin
I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
1
vote
0answers
20 views
Central Limit Theorem for Dependent Non-Identical Random Variables.
If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$.
How do we apply ...
2
votes
0answers
15 views
Multivariate Distribution Question?
If $(X,Y)$ have the following joint distribution:
$$f_{X,Y}(x,y) = \begin{cases}
2 f_X(x)f_Y(y) & \text{if }xy>0 \\[6pt]
0 & \text{otherwise}
\end{cases}
$$
where $f_X(·)$ and $f_Y(·)$ ...
1
vote
0answers
26 views
P.d.f of a discrete fourier transform of binary variables
Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
1
vote
1answer
24 views
Generalized chi distribution
Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied?
I know that its ...
5
votes
1answer
105 views
How was the normal distribution derived?
Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
0
votes
0answers
17 views
how to obtain the moments of skew-normal distribution?
the moment generating function of a skew normal distribution of random variable, z is defined as,
$$
M(t) = 2(e^{(t^2/2)})\Phi({{\delta}t)}
$$
where,
$\Phi$ refers to cumulative distribution function ...
1
vote
1answer
56 views
Derive the PDF of the log-normal distribution?
If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods:
(i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
5
votes
0answers
64 views
What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?
If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed.
What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
1
vote
1answer
21 views
Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
I am trying to derive Chi-square distribution. The random variale is
$$ U^2=\sum_{i=1}^k X_i^2 $$
where $X$ is a random variable with normal standard distribution.
What is the distribution of ...
0
votes
0answers
34 views
Can Bhattacharyya distance be greater than one?
I have two vectors, say $P$ and $Q$. I want to find the statistical overlap between two given that $P$ is my reference which I have modeled after Normal distribution and I have parameters for it. $Q$ ...
0
votes
0answers
23 views
Finding the joint distribution of 2 ratio of Gaussian random variables
Given independent normal random variables $X$, $Y$, and $Z$, I have the following ratios defined
$$
\begin{align}
r_1 &= \frac{x}{z}
\\
r_2 &= \frac{y}{z}
\end{align}
$$
The marginal ...
0
votes
2answers
29 views
probability normal distribution
A model for the movement of a stock supposes that if the present price of
the stock is s, then after one time period it will be either (1.012)s with
probability 0.52, or (0.99)s with probability ...
5
votes
0answers
52 views
Volume of the intersection of ellipsoids
How do I compute the volume of the intersection of two n-dimensional ellipsoids?
Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
0
votes
0answers
36 views
Inverse normal cdf of normal cdf with different covariance
Is there a analytical solution for the following transformation?
$$
y = \mathcal{N}^{-1} \left( \mathcal{N} \left( x | 0, \Sigma_1 \right) | 0, \Sigma_2 \right)
$$
where $\mathcal{N} \left( x | \mu, ...
0
votes
1answer
27 views
Normal distribution $\rho_{X,Y} = 0 \rightarrow X \bot Y$
Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$.
Can someone give a hint why this is true ?
1
vote
1answer
22 views
Independence of Combination of Normal Random Variables
$\newcommand{\Cov}{\operatorname{Cov}}$
I have a practice question I'm trying to answer in studying for an upcoming exam:
$X\sim N(0,1)$ and $Y\sim N(0,1)$ and I have $\rho(X,Y)=0.4$. Define a ...
0
votes
1answer
56 views
What does $E[{\bf{x}} {\bf{x}}^{T}]$ mean?
It's known that $E[{\bf{x}} {\bf{x}}^{T}]={\bf{\mu \mu}}^{T}+{\bf{\Sigma}}$ but I have seen a very similar identity using data points $\bf{x_{n}}$ and $\bf{x_{m}}$ sampled from a multivariate Gaussian ...
2
votes
1answer
69 views
What is the analytic expression for PDF of joint distribution of two Gaussian random vectors?
I know that if $X$ and $Y$ are random variables with respective PDFs,
$$
f_X(x) = \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left\{-\frac{\left(x-\mu_x\right)^2}{2\sigma_x^2}\right\} \\
f_Y(y) = ...
0
votes
1answer
24 views
What is the distribution of an unconditioned random variable knowing the conditional distribution?
I have two random variables $X$ and $Y$. I know that $Y$ can be approximated by a $N(\mu_1,\sigma_1^2)$ distribution (in particular $Y$ is not negative) and I also know that $X|Y \sim N(a+bY,c+dY)$ ...
1
vote
2answers
145 views
The correlation between two normal distribution
Let $X$ have the $N(0,1)$ distribution and let $a>0$, show that the random variable $Y$ given by
$$Y=\begin{cases}
X & \text{if }|X|<a\\[5pt]
-X &\text{if }|X|\geq a\;
\end{cases}$$
has ...
3
votes
1answer
142 views
Expectation value of $1/x$
Given a random variable $x$ which is assumed to follow a Gaussian distribution
$x \sim N( \mu, \sigma^2 )$ and $x$ is further known to be positive, I am interested in the following expectation value: ...
1
vote
1answer
49 views
Mentally Estimating the Normal CDF
More than once I have seen this sort of frustrating question on a Mathematics GRE practice test:
A fair die is tossed 360 times. The probability that a six comes up on 70 or more tosses is...
a) ...
0
votes
1answer
62 views
Distribution of Product of Random Variables with one being the normal distribution.
Let X and Z be independent, with $X\sim N(0,1)$, and with $\textbf{P}(Z=1)=\textbf{P}(Z=-1)=\frac{1}{2}$. Let $Y=XZ$ (i.e., Y is the product of X and Z).
(a) Prove that $Y\sim N(0,1)$.
(b) Prove ...
0
votes
1answer
43 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 ...
2
votes
0answers
102 views
Simplifying covariance matrices in distributions
In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
0
votes
0answers
41 views
Calculating Probability a Population is Normal Given a Sample
A lecturer is delivering a speech to 1000 people. She is meeting people in the audience and testing the hypothesis that the audience has a normal distribution of third sons (assuming that 200/1000 is ...
1
vote
1answer
92 views
Rayleigh distribution
I have this question from my statistical theory course:
A sniper shoots at a target. X and Y measure its deviation on the x and y axes. X and Y are independent and are distibuted normally with mean=0 ...
1
vote
1answer
40 views
What is the physical meaning of the output/ y -value of a normal distribution? (not the area under its curve)
Forgive me for my lack of knowledge regarding math terminology.
I'm learning basic statistics right now, and I can see pretty intuitively that the area under a normal distribution on a certain ...
0
votes
2answers
32 views
Algorithm for integral of standard distribution
I need help in producing random data that follows standard distribution.
Since it is to be used in a computer application, I would prefer an algorithm before a table.
So, this is what I need.
The ...
1
vote
1answer
56 views
About the differential entropies of well-known continuous distributions
Assume that the continuous random variable $X$ has a distribution (in a closed form expression) with differential entropy $h(X)$.
Q) Then, is it true for any continuous distribution that the ...
0
votes
1answer
100 views
Maxwell-Boltzmann velocity PDF to CDF
I need to draw from a Maxwell-Boltzmann velocity distribution to initialise a molecular dynamics simulation. I have the PDF but I'm having difficulty finding the correct CDF so that I can make random ...
1
vote
1answer
45 views
What is the distribution of empirical covariance between two independent normal distributions?
Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
1
vote
1answer
408 views
Variance for a product-normal distribution
I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution.
It's a strange distribution involving a delta ...
0
votes
1answer
30 views
Dividing data of a given distribution
Having a dataset with distribution $p$ (e.g. uniform or normal), if we divide the dataset into $n$ parts with equal size, is it valid to say that each part still has distribution $p$?
2
votes
2answers
84 views
What is the probability that two samples represent the same normal distribution?
Yes, it's a basic question. But, I have searched about 25 web pages for this and found only things that were irrelevant or incomprehensible. So I have indeed tried.
My question is: I have two ...
0
votes
0answers
32 views
Splitting a dataset with a given distribution?
Having a dataset with a given distribution $p$ (e.g. uniform or normal), if the dataset is randomly split into $n$ sub-datasets with equal size, is it valid to say that each sub-dataset still has ...
1
vote
1answer
57 views
Normal Distribution Identity
I have the following problem. I am reading the paper which uses this identity for a proof, but I can't see why or how to prove its true. Can you help me?
\begin{align}
\int_{x_{0}}^{\infty} e^{tx} ...
1
vote
2answers
50 views
Conditional Distributions and Probabilities
Suppose that $Y=A+\epsilon$ where $\epsilon$ is a RV and given some other random variable $\eta$ we have that:
$\epsilon|\eta$ ~ $N(\rho\eta,\sigma^2)$
Suppose I was asked to find $Pr(Y=y|\eta)$ ...
0
votes
3answers
112 views
Integrating the pdf of a normal distribution
I need to find the distribution of $Y=X_1+X_2$ where both $X_1$ and $X_2$ are normally distributed with $(\mu,\sigma^2)$.
So I'm looking for ...
0
votes
1answer
65 views
moment generating functions by integration
Let X~N(0,1)m find the moment generating function of $X^2$ using integration techniques.
I'm not sure exactly what this is asking me to do. Is $X^2$ just the pdf for the standard normal function ...
2
votes
0answers
28 views
Unknown result in probability theory relating CDF of any density to the CDF of normal distribution
There is apparently a result in probability theory saying:
If $A(z)$ is any cumulative distribution function, $\alpha(t)$, the corresponding characteristic function and $\Phi(z) = ...
0
votes
1answer
25 views
closed form for $p(B_1>x>B_2)$ where $[B_1, B_2]'$ follows a bivariate lognormal dist?
Is there a closed form for $p(B_1>x>B_2)$ where $[B_1, B_2]'$ follows a bivariate lognormal dist:
$$[B_1, B_2]' \sim \text{lognorm} (\boldsymbol \mu, \boldsymbol \Sigma)$$
where $\boldsymbol ...
2
votes
1answer
26 views
Multi-dimensional MLE Guassian
I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
2
votes
1answer
107 views
Conditional Expectations (Mainly an integral question)
Let $X_1$ and $X_2$ be two Random variables with a standard normal distribution, and the two variables are independent.
Find $E[X_1|X_1>X_2]$
My answer is far.
If we knew $X_2$, then the answer ...
0
votes
1answer
39 views
interpreting an expression involving two random variables
Consider a function $$g=E[\max(a+X,d+Y)]$$ where $a,d\in R$ and $X$ and $Y$ are independent and identically distributed standardized random variables with mean $\mu$, variance $\sigma^2$, continuous ...
0
votes
1answer
31 views
Computing expectation with n noisy sample?
Assume $\theta$~$N(0,\sigma^2)$, and we have $n$ realization of signals $s_i$, where $s_i$~$N(\theta,\sigma_i^2)$.
Now the question is: what is $E[\theta|s_1,s_2,\dots,s_n]$?
Thanks in advance.
0
votes
1answer
179 views
Application of Exponential Distribution
I'm in the process of developing a traffic simulation using Discrete Event Simulation approach.
So I have the core stuff working but I need to have some sort of distribution to tell the simulation ...
1
vote
1answer
291 views
Normal probability distribution with absolute value of X
Random variable X has a normal distribution N(30,5)
find $P(|X| > 25)$
Having this I started to solve it normal way:
$$P(|X| > 25) = 1 - P(|X| \le 25) $$
Now, normalize:
$$1-P(|X| \le 25) = ...
1
vote
0answers
54 views
Efficient calculation of the multivariate normal density function
The formula for the multivariate normal density function in the standard form contains $\Sigma^{-1}$ and the determinant of $\Sigma$, which are not very computationally friendly. Is it possible to ...
