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

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5
votes
1answer
109 views

Computing the inverse error function

I need a formula/series to compute the inverse error function, which is the inverse of $$ \operatorname{erf}(x) = \dfrac{2}{\sqrt{\pi}} \int\limits_{0}^{x} \mathrm{e}^{-t^2} \,\mathrm{d}t. $$ ...
0
votes
1answer
20 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 ...
2
votes
1answer
26 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 ...
1
vote
1answer
30 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 ...
0
votes
0answers
7 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 ...
0
votes
0answers
32 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
votes
1answer
55 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 + ...
0
votes
1answer
15 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
votes
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
votes
1answer
22 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
29 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 ...
-2
votes
0answers
16 views

CLT for Random Walks [on hold]

I am having trouble with the following problem. How do I show that it converges in distribution to a normal gaussian distribution? Along with the following questions of the problem. Click for image
-1
votes
2answers
30 views

Normal distribution, probability and modulus question [on hold]

Say $X$ is a random variable which is normally distributed with mean $0$ and variance $1$. How do I find $k$ such that $$\mathbb{P}(|X-k| < |X+k|) = 0.7$$
2
votes
3answers
47 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 ...
-5
votes
0answers
22 views

Multivariate Gaussian distribution [on hold]

I done parts (a) and (b) but I am stuck on part (c). I think the joint distribiution of R^2 is a chi squared r.v but I am not sure
0
votes
1answer
12 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 ...
1
vote
3answers
42 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 ...
1
vote
0answers
17 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
votes
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
votes
1answer
15 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 ...
2
votes
1answer
69 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') \ ...
0
votes
1answer
8 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
votes
0answers
22 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
456 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 ...
5
votes
3answers
111 views

Approximating the normal CDF for $0 \leq x \leq 7$

In answer http://stackoverflow.com/a/23119456/2421256, an approximation of the complementary normal CDF (ie $\frac{1}{\sqrt{2 \pi}} \int_x^{+\infty} e^{-\frac{t^2}{2}} dt$) was given for $0 \leq x ...
3
votes
1answer
42 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
vote
1answer
22 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: $y: N(2,3)$ $z: \chi^2(7 d.f.)$ ...
0
votes
1answer
504 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
18 views

Error function relation to the normal cumulative distribution function

A CDF for a normal standard is the following: $$N(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\phi^2/2} d\phi$$ I have the following relation in my notes which I am not very sure how they ...
0
votes
1answer
494 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
vote
1answer
21 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} ...
1
vote
0answers
45 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 ...
3
votes
0answers
46 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
vote
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, ...
1
vote
1answer
42 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 $Z_n \sim N(0,1)$ ?? ...
0
votes
1answer
22 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: ...
1
vote
0answers
30 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
votes
1answer
28 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 ...
1
vote
1answer
15 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
1answer
562 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 ...
0
votes
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} ...
-1
votes
0answers
11 views

Bivariate normal distribution (check)

I need to determine probability. Random variable X has a bivariate normal distribution with mean vector μ and covariance matrix Σ. $$X = {x_1 \choose x_2}, \mu = {-2 \choose 7}, \Sigma = ...
0
votes
0answers
18 views

How to find a good distribution

i have two distributions x1, x2 and when i plot x1/x2 i have a gaussian distribution but i ask for me is it is possible to have x1 and x2 gaussian ? because at the moment x1 and x2 looks like ...
0
votes
1answer
1k views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
2
votes
1answer
543 views

Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution

I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was ...
1
vote
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 ...
3
votes
0answers
83 views

Probability $P(X>Y,X>Z)$ for independent normal random variables $X$, $Y$, $Z$

There are several answers already given for working out the probability of one random variable being greater than another, but I can't make the leap to working out the probability of one random ...
0
votes
0answers
11 views

Composition of Binomial Distributions/Normal Approximation

I'm modeling as system as follows: $X_{0} = 1$ $X_{t+1} = X_{t} + Z_{t}$ $Z_{t} \sim Binomial(n=X_{t}, p)$ i.e. a composition of binomial distributions I'm interested in the variance of $X_{t}$ ...