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

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3
votes
1answer
21 views

Using the normal approximation, what is the $z$-value of a sample difference of $\hat p_1−\hat p_2=−0.18$? [closed]

A few concepts from my textbook that I do not understand: If the Random, Normal, and Independent conditions are met, then is it true that $\hat p_1+2\hat p_2$ is approximately normally ...
1
vote
3answers
59 views

Find $f$ and $g$ such that $\Phi(x)f(y)=g(x+y)$

Let $\Phi: \mathbb{R} \to [0,1]$ define the standard normal cdf function. I am trying to find some functions $f$ and $g$ -- if they exist -- such that $$\Phi(x)f(y)=g(x+y)$$ for all $(x,y)\in \mathbb{...
0
votes
0answers
48 views

How to show this function is increasing. Related to bivariate normal distribution.

How to show this function is increasing function: Define $f(z)=\frac{T'(z)}{T(z)}-\frac{g'(z)}{g(z)}$, where $T(z)=\int_{-\infty}^z \Phi(x) -\Phi(\frac{x}{\alpha})\, dx$ and $g(z)=Pr(Zi<z,Zt>\...
2
votes
1answer
57 views

On expectation of maximum of gaussians

Let $X_1,\ldots,X_n$ be i.i.d $\mathcal{N}(0,1)$ random variables. I am trying to prove that \begin{align} (a)\ \ \mathbb{E} \left[ \max_{i}X_i\right] & \asymp\mathbb{E} \left[ \max_{i}|X_i|\...
0
votes
1answer
43 views

Expected value of ratio of normal CDFs

I am trying to compute the expected value of the ratio of two normal CDFs. Specifically, I like to compute the expected value of $\Phi(X+Y)/\Phi(X)$ where $X$ and $Y$ are independent normally ...
0
votes
1answer
23 views

Law of total expectation not converge?

So the question asks: Let $X, Y$ be random variables, with joint probability density function: $$f_{X,Y}(x,y) = \left\{ \begin{array}{ 1 l } 0.25ye^{-y} & \mbox{if $0≤|x|≤y$}\\ 0 & \...
0
votes
0answers
24 views

integral of heteroskedastic Gaussian

For a Bayesian analysis I need to solve several integrals of the following kind. Let's start with the simplest 1-D form: $$ \mathcal{I}_k = \int_{-\infty}^\infty s^k \mathcal{N}\left(s|x,\sigma^2(s)\...
0
votes
2answers
50 views

Find $P(Z>1.8)$ and $P(X>4)$ when $Z$ is normal RV and $X$ is a binomial RV.

Question 1 If $Z\sim N(0,1)$, Find a) $P(Z>1.8)$ b) $P(Z >-2.46)$ c) $P(Z>2.589)$ d) $P(Z<-0.725)$ e) $P(Z<-1.63)$ f) $P(Z>-0.65)$ My Solutions: $P(Z>1.8) = 0.9461$...
1
vote
1answer
29 views

Probability generating function of negative binomial distribution proof

So the textbooks says: Let $X_r$ ~ $NB(r,p)$. We could use the probability generating functions to prove that $G_{X_r} (s) = (\frac{ps}{1-(1-p)s})^r$: Let $X$ have the Geometric distribution ...
1
vote
1answer
23 views

Multi-derivative containing standard normal CDF

I've got a question about the following multi-derivative $$\frac{\text{d}^m}{\text{d}a^m} \left(e^{-2\mu a}\Phi \left(\frac{a-\mu u}{\sqrt{u}}\right)\right),$$ where $m> 0$ is an integer , $\mu&...
1
vote
0answers
13 views

Multivariate distribution with the same kurtosis as normal distribution

Good morning. I am writing a thesis about testing multivariate normality. I would like to do a comparison of power of some tests against given alternatives based on Monte Carlo simulations. I have a ...
0
votes
0answers
13 views

Formula for linear range to normal distribution with arbitrary peak

Let's say I have a system that produces percentage probability scores. These scores can, being percentages, range from 0 to 100. For the processes that produce my score, results within the entire ...
0
votes
0answers
50 views

Density function of $\sqrt{(\mathcal{N}_1(\mu,\sigma^2))^2+(\mathcal{N}_2(\mu,\sigma^2))^2}$ (Random walk)

I have 2D random walk and I would like to find out what distance I will travel after 200 steps. So I introduce two random variables $Z^{(200)}_x$ and $Z^{(200)}_y$ which tell me probabilities of my $x$...
0
votes
1answer
21 views

Frequency pdf distribuiton

The oscillatory frequency is defined by $F=1/T$. Given that T has a normal distribution $N(µ_t,\sigma_t$) how do I calculate the frequency distribution based on period $µ_t,\sigma_t$ ? I assume that $...
0
votes
0answers
65 views

Covariance and integral of multiplied Gaussians

There is a 2-dimensional Gaussian $G(x, y)$ $$ G(x, y) = c \exp \left(-\frac{(x - \mu_x)^2}{2\sigma_x^2}\right) \exp \left(- \frac{((y - \mu_y) - a(x - \mu_x) )^2}{2\sigma_y^2} \right) $$ where $c$ is ...
1
vote
0answers
41 views

How to do the convolution of a normal distribution with a truncated exponential distribution?

I have a random variable $A$ with $A = B + C$, where $B$ is a normal distribution with the usual range $(-\infty , +\infty)$ and $C$ is a truncated exponential distribution of range $(a,b)$. How to ...
1
vote
1answer
64 views

Find the probability that a randomly selected box will contain more than $15$ of the tubes with length longer than $2.1$ cm.

Problem: A manufacturer produces a certain type of tubes that has a length of $2.1$ cm. Assume that the distribution of this type of tubes is normal with mean $2.15$ cm and standard deviation $0....
0
votes
1answer
71 views

Normal approxiamation to Binomial distribution

A company has $200$ employees. Assume that each employee invites $2$ guests, independently, to attend a promotion seminar with a probability of $0.8$. How many seats should be provided if the company ...
2
votes
0answers
26 views

Gaussian and binary probability random variables numerical reconstruction

in my probability class I was given this question dealing with MATLAB code the purpose of which is to create and re estimate the random variables Z1, Z2, which reads as follows: rng('default') ...
0
votes
1answer
70 views

Gaussian random variables independent iff statement

Let $X_1, \ldots , X_n:\Omega \to \mathbb{R}$ be random variables such that $(X_1, \ldots , X_n)$ is a Gaussian vector in $\mathbb{R}^n$. I want to show that $X_1$ and $\sigma(X_2, \ldots , X_n)$ are ...
0
votes
1answer
38 views

Problem understanding how $P(X\leq x)=P\left (Z\leq \frac{x-\mu}{\sigma}\right )=\Phi\left (\frac{x-\mu}{\sigma}\right )$

Ok so we have the cumulative function for the standard normal distribution: $$\Phi(z)=\frac{1}{\sqrt{2\pi}}\int^{z}_{-\infty}e^{-t^2/2}\;dt$$ and the cumulative function for the general normal ...
1
vote
1answer
125 views

Calculating the PPV and NPV using Bayes' Theorem [closed]

A kit manufacturer markets a diagnostic kit for use in mass screening for Thyroid disease. The diagnostic kit has the following specification: Diseased Population: Has response variable of $N(11,64)$...
1
vote
1answer
39 views

Expected value after a chain of events?

my problem is the following. I have a scalar variable, which changes with a certain noise every step: Step 0: $x_0 = \nu_0$, Step 1: $x_1 = s\cdot x_0 + \nu_1$, ... Step i: $x_i = s\cdot x_{i-1} + ...
1
vote
0answers
23 views

Distribution of squared multivariate normal random variable

Let $W\sim MVN(\mu, \Sigma)$, here $W$ and $\mu$ are $k\times 1$ vector and $\Sigma$ is $k\times k$ symmetric matrix. And the diagonal elements of $\Sigma$ are equal to one. For this multivariate ...
0
votes
0answers
29 views

Multi-derivative of standard normal CDF

I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , $\mu>0$,...
0
votes
0answers
45 views

A function of bivariate normal distribution. How to show the function is increasing and convex?

How to show this function is convex, increasing function: Define $f(z)=\frac{T(z)}{g(z)}$, where $T(z)=\int_{-\infty}^z \Phi(x)\, dx-\alpha\int_{-\infty}^{\frac{z}{\alpha}} \Phi(x)\, dx$ and $g(z)=...
2
votes
0answers
36 views

Question on Poisson distribution approaching the normal distribution.

Suppose $X$ is a Poisson$(\lambda)$ random variable. I have already shown as part of the question that $\sqrt{\lambda}\Pr(X=\lambda+x\lambda) = \frac{1}{\sqrt{2\pi}}\exp^{\frac{-x^2}{2}}$ but I need ...
0
votes
0answers
27 views

Gamma distribution of $Y^2 \sim Γ(0.5,0.5)$

So the question asks: Let $X\sim Γ (s,λ )$ be a random variable distributed according to a gamma distribution (with $s$, $λ > 0$). Suppose $Y$ is a standard normal random variable. Show that $Y^2 \...
1
vote
1answer
27 views

probability of a standard normal distribution

I haven't been able to make the sense of the following expression and use the table for standard normal distribution. z.10
1
vote
0answers
36 views

Convergence of normal distributed random variables

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables with the property that $N(m_n, \sigma_n^2)$. Prove that if $X_n$ converges in distribution to $X$ then $X \sim N(m,\sigma^2)$ where $m =...
1
vote
1answer
27 views

Upper bound inequality cumulative normal distribution

According to this post, I found for $X \sim N(0,1)$, $x > 0$ the result that \begin{align} \frac{1}{\sqrt{2\pi}}\big(\frac{1}{x}-\frac{1}{x^3}\big)e^{-\frac{x^2}{2}} \leq P(X>x) \leq \frac{1}{\...
0
votes
0answers
19 views

Solve an integral of a Multinomial with Gaussian prior

I have a Multinomial whose parameters evolve over time following a random walk with diagonal covariance. I have a sample of the multinomial for each time. $$p(X_t) = Multinomial(\theta_t)$$ $\...
0
votes
1answer
16 views

Finding Z in a normal distribution

I am trying to answer the following: Let X be a continuous random variable with normal distribution with parameters μ = 3 and σ = .2. Find x so that the following holds. P(X > x) = 2.5% I tried ...
0
votes
0answers
29 views

Distribution of sample variance

I have a problem and I am really not sure how to process it. We have 10 samples of 25 variables X~N(50,25). How many of these samples will have a variance of maximum 101? $W = \frac{(n-1)S^2}{...
0
votes
1answer
32 views

Central Limit Theorem Question Relating to Sample Mean

I have the following question: Use the Central Limit Theorem to show at least how big $n$ must be so that the following is true: $P(|\bar{X}-\mu| \le 1) \ge 90\%$ with mean $= 7$ and standard ...
0
votes
0answers
29 views

A sequence of iid standard normal random variable

Let $X_1, X_2, ...$ be iid and have standard normal distribution. Let $b_n$ be a sequence of real number such that $$\mathbb{P}(X_1 > b_n) = n^{-1}.$$ It can be shown that $$\mathbb{P}(b_n(\max_{1 \...
0
votes
0answers
10 views

conditional expectation and variance multivariate normal (n>2)

There is a lot of documentations for the bivariate normal distribution but I was wondering what would be the formulas for the conditional expectation and conditional variance of a multivariate ...
2
votes
2answers
78 views

How can I evaluate $\int_{-\infty}^{\infty} e^{-x^2} dx$ without using polar coordinates [duplicate]

I know from probability class that the area under the bell curve $e^{-x^2}$ is $\sqrt{\pi}$. I would like to be able to verify this, so in other words, solve this integral: $$\int_{-\infty}^{\infty} e^...
0
votes
1answer
32 views

How to find specific percentage in a normal distribution

Here is the exercise question I'm trying to answer: A certain fast food outlet produces hamburgers with weights that are normally distributed with mean $\mu = 990g$ and standard deviation $\...
0
votes
0answers
27 views

Commute expected value and argmax for a noisy function with known argmax

I have the following question in Given a random sequence $X_t, t\in[1,T]$, with $X_t = \varepsilon_t$ for $t=1,\ldots,d$ and $X_t = \mu + \varepsilon_t$ for $t=d+1,\ldots,T$ and $\mu>0,\...
1
vote
2answers
59 views

How many positive integer solutions are there to $x_1+x_2+x_3+x_4<100?$

How many positive integer solutions are there to $x_1+x_2+x_3+x_4<100?$ I know how to approach the problem if it were How many positive integer solutions are there to $x_1+x_2+x_3+x_4=100$, it ...
0
votes
0answers
37 views

Division of Normal and Poisson Distribution

I am trying to understand the following: If $X$ is normally distributed and $Y$ is distributed according to the Poisson distribution, how to find out which distribution $Z=X/Y^2$ has? Is there any ...
1
vote
0answers
28 views

A closed-form formula for Cov(X,Y) when X and Y are normal random vectors?

I cannot figure out one step given in my textbook, [1] Mixed Models. Theory and Applications by E. Demidenko. I study the Linear Mixed Effects (LME) model in the following form: $$\mathbf{y_i}=\...
0
votes
0answers
46 views

Bivariate Normal Distribution of a Function of a Random Variable

Let X and I be two independent normally distributed random variables. This implies that X and I are jointly normally distributed. Let S=X+I As far as I know, X and S are jointly normal. How could I ...
0
votes
0answers
88 views

Proving chernoff bound (erfc function)

I have this question. By considering the probability that 2 independent, standard normal random variables, $x_1$ and $x_2$, lie within the square: $\{(x_1,x_2)||x_1|<x,|x_2|<x\}$, prove the ...
0
votes
3answers
46 views

Inverse standard normal CDF

We are dealing with a standard normal random variable. We have $$\Phi(c) = 0.8$$ where $c$ is just some arbitrary number and $\Phi$ is just the usual notation for the CDF of a standard normal ...
3
votes
2answers
114 views

Correlated joint normal distribution: calculating a probability

Given $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \right) $$ $Y = Z\sqrt{1-\rho^2} + \rho X$ And $$ f_{XZ}(x,z) = \frac{1}{2\pi } \exp \...
0
votes
1answer
10 views

$\alpha$ in a normal confidence interval

Related to: Deriving the confidence interval $P(-\Phi^{-1}_\alpha < X < \Phi^{-1}_\alpha) = 1-2\alpha$ I'm not sure why, but I'm having some trouble with the definitions here. $$P\sim\...
0
votes
1answer
27 views

Deriving the confidence interval $P(-\Phi^{-1}_\alpha < X < \Phi^{-1}_\alpha) = 1-2\alpha$

I'm trying to derive the confidence interval for the standard normal distribution. Let $P\sim\mathcal N(0,1)$ \begin{align} &P(-\Phi^{-1}_\alpha < X < \Phi^{-1}_\alpha)&&&...
4
votes
2answers
72 views

Normalized partial sums of normal random variables are dense in $\mathbb{R}$

I came across an interesting result appearing as an exercise in some lecture notes I'm reading. Suppose $X_{1},X_{2},...$ are IID $N\left(0,1\right)$ RVs all defined on $\left(\Omega,\mathcal{F},\...