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

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1answer
36 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
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1answer
24 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
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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
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0answers
47 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
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1answer
38 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
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0answers
38 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
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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 ...
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2answers
34 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
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2answers
35 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 ...
0
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1answer
28 views

Formula for probability of being $\epsilon$ within the mean.

It should be possible to restate that as $P(\mu-\sigma \Phi^{-1}(\frac{p+1}{2})\leq X\leq \mu+\sigma \Phi^{-1}(\frac{p+1}{2}))=p$. In this answer, it says: For a normal distribution, the ...
0
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1answer
20 views

What is the radius of a “Gaussian” sphere such that approx. all the population lie within?

Let $\mathbf{X}\in\Bbb{R}^n$ be a random vector distributed normally, i.e. $\mathbf{X}\sim\mathcal{N}(\mathbf{0},\sigma^2I_n)$, where $\sigma>0$ is the standard deviation and $I_n$ is the identity ...
2
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3answers
50 views

How is the entropy of the normal distribution derived?

Wikipedia says the entropy of the normal distribution is $\frac{1}2 \ln(2\pi e\sigma^2)$ I could not find any proof for that, though. I found some proofs that show that the maximum entropy resembles ...
1
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0answers
17 views

Problem on Mann Whitney U test statistic

Let $X_1, X_2, \ldots, X_m \sim N(\mu_1, \Sigma)$ and $Y_1, Y_2, \ldots, Y_n \sim N(\mu_2, \Sigma)$. (Here, $\Sigma$ is the variance-covariance matrix of the 2 multivariate Normal distributions ...
0
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0answers
28 views

On the derivation of the Cauchy Distribution

I am currently studying from this video lecture series and the professor here goes over the derivation for the Cauchy distribution. I am able to follow most of it except for one minor part. Part of ...
0
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1answer
52 views

Estimator of expectation value for standard normal distribution

In the case of a standard normal distribution, I just read that a good estimator for E[f(x)] is $\frac{1}{M}\sum_{i=1}^M f(X_i)$ (where each $X_i$ is standard normally distributed and independent). ...
2
votes
1answer
67 views

What is the pdf of sum of log-normal and normal distribution?

The question goes like this: $Z = X+Y$; where $X$ is Log-normal Random variable with parameters - $\mu = 0 \quad \sigma^2= 1$, $Y$ is Gaussian Random variable with $\mu= 0\quad \sigma^2= 1$ What is ...
0
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0answers
8 views

Operations on two normal distributions using order statistics

$G(x)$ is a Normal distribution with mean $\mu$ and standard deviation $\sigma$. I observe realization of $X$ which are a function of $s$. The distribution $F(s)$ is found as the root (between 0 and ...
1
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1answer
43 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as $e^{m+\frac{1}{2}...
0
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1answer
43 views

Nonstandard normal distribution

I want to understand how to prove results regarding the relationship between the standard and nonstandard normal distributions. In other words, I want to prove the results regarding how to use z ...
0
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1answer
59 views

Finding the distribution from the moment generating function

Let $X_1, X_2, · · · , X_n$ be a random sample of size n from a geometric distribution withpmf $f(x) = 0.75 · 0.25^{ x-1} , x = 1, 2, 3, ··· .$ (a) Find the mgf $M_{Y_n} (t)$ of $Y_n = X_1 + ...
2
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1answer
35 views

Distribution of residual term in regression.

In regression analysis for classical linear regression model the residual term is independent of x and y and normally distributed and it is a random variable but i found somewhere written u~N and u~...
0
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1answer
19 views

Distribution Theory - bivariate normal distribution

Question: Let X and Y have a bivariate normal distribution with E(X) = 5, E(Y ) = −2, var(X) = 4,var(Y ) = 9, and cov(X, Y ) = −3. U and V are defined as U = 3X + 4Y and V = 5X − 6Y .Determine the ...
0
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1answer
17 views

52% of people want to ban smoking Use the normal approximation to estimate that over half of a sample size $n$ support the the ban

52% of people want to ban smoking. Use the normal approximation to estimate that more than half of a given sample size $n$ support the the ban. q=1-0.52=0.48 For $n=11, 101, 1001$ Are these steps ...
0
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1answer
26 views

Let $ 0 \lt \alpha \lt 1$. $z_a$ is a solution to $\Phi(z_a)=\alpha $.

Let $ 0 \lt \alpha \lt 1 $. $z_a$ is a solution to $\Phi(z_a)=\alpha $. 1.) What is the relation between $z_a$ and $z_{(1-a)}$ 2.) Find $z_a$ (with an error that does not exceed 0.01) for the ...
0
votes
2answers
31 views

Central limit theorem on packs of variables

I'm trying to solve the following exercise: Let $\mu$ be a probability distribution on $\mathbb{R}$ having second moment $\sigma^2<\infty$ such that if $X$ and $Y$ are independent with law $\mu$...
3
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3answers
50 views

Let $X\sim N(3,4)$. Find $\mathbb{P}(X<7)$, $\mathbb{P}(X \ge 9)$

Let $X \sim N(3,4)$. Find $\mathbb{P}(X\lt7)$, $\mathbb{P}(X \ge 9)$, and $\mathbb{P}(|x-3|\lt 2) $ Okay lets figure out the PDF. $\mu=3$, $\sigma=4$. $$f(X)= \frac{e^\left(\frac{-(x-\mu)^2}{2 \...
-1
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2answers
38 views

Normal distribution, probability and modulus question [closed]

Say $X$ is a random variable which is normally distributed with mean $0$ and variance $1$. How do I find $k$ such that $$\mathbb{P}(|X-k| < |X+k|) = 0.7$$
0
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1answer
13 views

Normal Distrubution Question - How many components are defective and acceptable?

A component is defective if oversized. A sample of 460 components produced by a machine have a mean size of 7.2 cm and a standard deviation of 0.12 cm. The maximum size of an acceptable component is 7....
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3answers
46 views

Calculate $E(X^{2n})$ where $X$ is normal (0,1)

I need help proving the following: Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that: $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$ I've ...
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0answers
18 views

Normal Distrubition Question - How many wires will meet specifications?

Wires manufactured for use in a certain computer system are specified to have resistances between 0.12 ohm and 0.14 ohm, the actual measured resistances of the wires produced by company A have a ...
0
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0answers
27 views

Ratio of two normal random variables with the same mean and same standard deviation

I would like to compute the probability density function of $Z = \dfrac{X}{Y}$ with $X$ and $Y$ following a non-standard normal distribution with the same parameters (same mean and variance).
0
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1answer
18 views

GRE Quantitative problem on distributions

I was doing some problems on this .Can some one please help me with the following: Here the given answer is that quantity B is grater than Quantity A. How is this obtained? Do we know anything ...
0
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1answer
13 views

Lognormal distribution inverse equivalent

In Lognormal distribution if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Is there inverse equivalent to lognormal distribution where Y = exp(X) has a ...
0
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0answers
39 views

Gaussian processes and bias

I would like to simulate two Gaussian processes along a time grid. Ideally, I would like to see the average of the samples, for each grid point, to be close to the mean. Using the antithetic method, I ...
2
votes
1answer
74 views

Conditional Expectation of the minimum of two identical log-normal distributions

I'd like to compute the closed form mean of the minimum of two truncated log-normal distribution (on another interval than its truncation). I have: $\int_{a}^{\infty} \int_{a}^{\infty} min(v, v') \ ...
3
votes
1answer
43 views

Characterization of Normal RVs by uni variate version?

If $X$ is a symmetric $n$-dimensional random vector with mean $0$ then is it true that: \begin{align*} & X \text{ follows a multivariate normal law} \\ & \text{iff} \\ & \|X\| \text{is a ...
1
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1answer
25 views

Find the critical value of given statistical problem, t-distribution

My solution doesn't match the one given in my course, however I can't quite see what I've done wrong. Can someone give me a heads-up? Problem Given the following: $y: N(2,3)$ $z: \chi^2(7 d.f.)$ ...
2
votes
1answer
33 views

bayesian posterior of truncated normal distribution with uniform prior

Let $N_T(\mu,\sigma)$ be a truncated normal distribution with support on $[0,1]$. Draw $x \sim N_T(\mu,\sigma)$ (What I want to model is, I have a unknown quantity $\mu \in [0,1]$, but I only ...
2
votes
0answers
21 views

$(X_n)_{n\in\mathbb{N}} $ independent with standard Gaussian distribution

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables, each with standard Gaussian distribution. For a given $K>0$, prove that: $$\lim_{n\to\infty} \frac{1}{n}\log{P\left(\...
3
votes
0answers
49 views

Find $P(B_3>0,B_6>0)$ where $(B_t)$ is a Brownian motion

Suppose that $B_{t}$ is a standard Brownian Motion. What is the probability that both $B_{3}$ and $B_{6}$ take positive values? This is what I've tried but then I get stuck and I'm not sure how to ...
1
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1answer
22 views

Derivation of a property of standard Wiener processes

I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: $W(t) = 0$ $W(t) - W(s) \sim N(0, t-...
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0answers
33 views

What distribution results from drawing random numbers whose upper bound is normally distributed?

I have a normal distribution $N$ with $μ=U/2$ and $σ=U/12$ (an approximation of the Irwin-Hall distribution) which has been bounded and normalized to $[0,U]$. I will now repeatedly generate random ...
0
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1answer
55 views

Distribution of the product of two lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas. Consider the corresponding log-normal random variables: $...
0
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1answer
17 views
1
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0answers
33 views

Find the characteristic function of Y where Y|X=x $\in N(0, x)$ with X $\in Po(\lambda)$ [closed]

In particular $Var(Y|X=x)=x$. I think the solution should be the characteristic function for the $N(0,\lambda)$ distribution i.e. $\varphi_y(t)=e^{-\frac{1}{2}t^2\lambda}$ but I can't figure out why ...
0
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1answer
32 views

Variance of a Cumulative Distribution Function of Normal Distribution

Suppose, $X\sim N(\mu,\sigma^2)$. Can anyone help in finding the following : $\text{Var }\Phi(X)$ ? Here, $\Phi(x)$ is the "Cumulative Distribution Function" of the above-mentioned normal ...
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1answer
48 views

Why $Z_n$ is normally distributed?

We know $\epsilon_n \sim N(0,1)$, and $$Z_n = \frac {\mu_n^T(I-M_n)\epsilon_n} {\sqrt {\mu_n^T(I-M_n)\mu_n}},$$ where $M_n=X_n(X_n^TX_n)^{-1} X_n^T$, $\mu_n=X_n\beta_n$. Why $Z_n \sim N(0,1)$ ?? ...
1
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1answer
17 views

Link between conditional characteristic function and conditional density

Let $X$ and $Y$ be random variables (real-valued). I define $$E[e^{i\theta X}\mid\sigma(Y)] =: g(Y,\theta)$$ Suppose that $g(Y,\theta) = e^{i\theta Y}e^{-\frac{1}{2}\theta^2}$. Can I then say that ...
1
vote
0answers
47 views

Integrating a Random Variable and establishing the maximum of a related function

Frequency Regulation of a Power Grid I have a battery that is used to regulate the frequency of a power grid. That is, as the grid frequency varies about it’s ideal value, $f_{nom}$, the battery ...
0
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0answers
16 views

Probability of gaussian random variables lying in a certain order

I have two independent gaussians and a known constant: $$ \begin{align} X_1 &\sim \mathcal{N}(\mu_1, \sigma_1^2) \\ X_2 &\sim \mathcal{N}(\mu_2, \sigma_2^2) \\ c &\in \mathbb{R} \end{...