# Tagged Questions

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

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### The probability that the ratio of two independent standard normal variables is less than $1$

Let the independent random variables $X,Y\sim N(0,1)$. Prove that $P(X/Y < 1) = 3/4.$ Could anyone help me prove this analytically? Thanks. Progress: My first thought was to integrate the joint ...
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### Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
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### Basically Normal Dist question

I'm a little rusty on my probability and would appreciate any help. I think I have done the bulk of the work already anyway, but my question is: If $X \sim LN(1,2)$ find $P(X>1)$ $X$ being ...
I am having trouble deriving the mean for a multivariate normal for $\mathbf{x} \sim \mathbb{N}(\mathbf{m},\Sigma)$: $$\mathbb{E}[\mathbf{x}]= \int_{R^d} \mathbf{x} ... 1answer 130 views ### Bayesian Updating with 1 Signal but 2 Unknowns Suppose I have an unknown variable X_i = \alpha_i + \beta_i where \alpha is one of 2 different values {{\alpha_1, \alpha_2}} such that \alpha = \alpha_1 with probability p_1 and \beta is ... 1answer 31 views ### Distribution with density x^2\operatorname{exp}\{-x^2/2\} I came across the probability distribution with density$$ f(x)=\sqrt{\frac{2}{\pi}}\,x^2\,\mathrm{e}^{-\frac{x^2}{2}},\quad x\geqslant 0. $$Is this distribution known under a certain name? I only ... 1answer 32 views ### The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ... 1answer 207 views ### Simple calculations of mean, standard deviation, and probability You are a successful entrepreneur that has developed a new sustainable product that is manufactured through a standard production process. As part of this process, the product goes through quality ... 2answers 677 views ### Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve? a) P(0 < Z < 2.17) b) P(-2.50 < Z < ... 0answers 20 views ### Normal distribution tables - right or left? Are the probabilities in normal distribution tables given typically to the right or left of the Z score? One such text I am reading says to the right. However, in my lecturer's exercises, I ... 0answers 20 views ### Approximate distribution of product of N normal i.i.d.? Given N>30 i.i.d. X\approx\mathcal{N}(\mu_X,\sigma_X^2), looking for: accurate closed form distribution approximation of Y=\prod_{n=1}^{N}{X} asymptotic normal approximation of same ... 1answer 77 views ### Generate a uniform distribution from n coin flips I'm making a computer game and I've reduced the problem into something simple: How can I show the player the number of heads he "tossed" given some number of coins = n? Naive expected value is ... 0answers 41 views ### Moments of Multivariate Normal Distributions I have two questions. Suppose we have two multivariate normal distributions X \sim N(\mu,\Sigma) and Y\sim N(c\mu,\Sigma) where 0<c<1 is a constant, \mu is a vector and \Sigma is a ... 1answer 34 views ### Find probability given a binomial and a normal distribution X~Bin(n,p),Y_n~N(μ,\sigma^2) Where X is the number of trials taking place, and Y_n is the amount of time the nth trial takes (independent of other trials). Z is a new random variable ... 2answers 26 views ### Calculating a normal distribution with a sample size? the sample of n=25 is what is throwing me off. I have no clue what to do with it. Given a normal distribution with \mu=101, \sigma=25, and given you select of n=25 A.) P(\overline{X} ... 0answers 104 views ### Approximating the integral of the exponential of a quadratic function An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where c=b^2/(4a)+\ln(-a/\pi)/2 and ... 0answers 104 views ### How to construct a two sided confidence interval? A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data are shown below. 5.21 5.28 5.29 5.27 ... 0answers 38 views ### Finding the conditional distribution of a normal RV given another normal RV I'm having trouble with this question from a past qualifying exam: Question Suppose Z \sim N(\mu,\sigma^{2}), W \sim N(0,1) and V \sim N(0,1) are mutually independent normal random variables. ... 0answers 32 views ### Can a mixture of normals be a constant? Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane? ... 1answer 31 views ### Square Matrix Algebra - help please! I am stuck on a problem in matrix algebra and I would be happy if someone could help me. Given a square matrix with dimensions "p" given that \textbf{x} \sim N(\mu,\Sigma) [multivariate ... 1answer 50 views ### How to interpret the covariance matrix of Brownian motion I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion B_t is Gaussian Process, i.e. for all 0 \leq t1 \leq \cdots \leq t_k the random variable Z = ... 0answers 56 views ### If X|Y and Y are both normal, is X|Y>y normal as well? Consider two random variables, X and Y, with the following properties: X|Y\sim N(Y,s^2) and Y\sim N(\mu,\sigma^2). Does X|Y>y follow a normal distribution as well? If so, what are its ... 2answers 35 views ### correlation between \sum_{i=1}^{98}X_i and \sum_{i=3}^{100}X_i Let X_1,...,X_{100} be iid N(0,1) random variables. The correlation between \sum\limits_{i=1}^{98}X_i and \sum\limits_{i=3}^{100}X_i is equal to (A) 0 (B) \dfrac{96}{98} (C) ... 1answer 57 views ### Central Limit Theorem sample vs population I need help in the setup of this problem. I'm sure that I'm making this far more complicated than what it actually needs to be. "An anthropologist wishes to estimate the average height of men for a ... 0answers 22 views ### If I approximate a Bionimial distribution with a Normal Distribution am I still allowed to use Binomial's equation for Variance? If I approximate a Binomial distribution with a Normal Distribution am I still allowed to use Binomial 's equation for Variance? So am I still allowed to use this: Var(x) = np(1 − p) While still ... 0answers 96 views ### Expected value of a function of truncated normal I need to find the expected value of the following type of an expression:$$\mathbb{E}[\frac{1-\alpha}{1-\alpha-\frac{X}{\beta}}]where \alpha and \beta are constants and X is a random ... 0answers 69 views ### Question Relating to the Central Limit Theorem I have the following question: Suppose X_1,X_2, \ldots, X_{12} are identical independent uniform random variables on [0,1]. Let the sample mean(lets call it m) = \frac{1}{12}(X_1 + X_2 + ... 0answers 23 views ### Distribution of unknown, given system of equations Suppose we have an unknown real x. We want to give an approximation of x by measuring the distance between s_i and x, for i = 1,2,3. The position of each s_i is distributed with mean p_i ... 1answer 25 views ### Calculate E(X^2) of random variable X ~ N(3,4) I need to find E(X^2) of random variable X ~ N(3,4). I can use the simple way: E(X^2) = \int_{-\infty}^{\infty} x^2 \cdot f(x) dx, in this case f(x) = normal \space distribution ... 0answers 22 views ### Poisson distribution with normal informative priors I'm Jia, a student of economics and finance. I was wondering if someone could help in understanding this problem. I've just started to attend a new course "Financial and nonlinear econometrics" and ... 1answer 37 views ### Distribution of two independent standard normals Suppose that X and Y are distributed as independent Standard Normals. Find the distribution of (X-Y, X+Y). Isn't the case for X-Y elementary? Since they are both standard normals, this ... 1answer 44 views ### Chi distribution and sample variance Suppose that the height (in cm) of randomly selected male is distributed according to normal distribution with parameters \mu = 175 and \sigma = 5. We pick a simple random sample of size ... 3answers 49 views ### Expected value of random variable X ~ N(170, 25) Here's a question: Person's height in CMs is a random variable X ~ N(170, 25). Door's height is 180 cm. What is the expected value of number of people that can enter the door until the first ... 1answer 36 views ### Seminorms in distribution theory are norms, right? In distribution theory the seminorms that you use there p_m( \phi) := \max_{|\alpha| \le m} \sup_{x \in \Omega} |(\partial^{\alpha}(\phi) (x)|, \phi \in C_c^{\infty}(\Omega). Those guys are norms ... 1answer 85 views ### Random Gaussian variable raised to arbitrary power Given x that follows a distribution P(x)=e^{\frac{-x^2}{2\sigma^2}} i.e. a random Gaussian variable, can I say anything about the distribution of x^n for fixed n? Specifically, is there ever ... 0answers 30 views ### sampling from a multivariate guassian: intuition behind using cholesky decomposition I'm trying to understand how sampling from a multivariate gaussian works and why the cholesky decomposition is a way to do it. Let's say we have a 25 dimensional multivariate with a 25x25 covariance ... 1answer 65 views ### Smallest n to align sample mean with population mean There's a question in my book that I just do not understand. This is it in its entirety: Let  \bar{X}  be the sample mean of a random sample of size  n  from a normal distribution with a variance ... 1answer 55 views ### On the notation of normal distribution I saw in the Finnish matriculation examination solutions the sentence If X has the distribution N(100,15), Z=\frac{X-100}{15} has the distribution N(0,1). How one can memorize this? I mean ... 1answer 122 views ### Sample standard deviation and population standard deviation The average temperature of a particular tropical island is normally distributed with a mean of 74 degrees and a variance of 9 degrees (a) If a random sample of 16 days has been taken, what is the ... 2answers 50 views ### How to set up normal approximation for binomial In a particular school, 25% of first grade students do not enjoy reading. 22% of second graders do not enjoy reading. A random sample is taken of 100 first grade students, and another independent ... 0answers 17 views ### Statistics, distributions and graphing Lets say that you have data containing one variable - the waiting time between each car passing from 1200 hours... so you have something like 23, 54, 26, 8, 2, 59 etc what is the best way to analyse ... 1answer 28 views ### Solving for an unknown \mu in a probability problem involving normal random variables. (a): P[X < 355] = P[Z < \frac{355 - 360}{4}] = P[Z < -1.25] = 1 - \Phi[1.25] = .1056. Part (a) is simple, but I included it because I was not sure if I should somehow use it to solve ... 0answers 53 views ### How to obtain the pdf of this transformation of normal random variables? Given the following: Let r = ab + n, where a, b, and n are independent zero-mean Gaussian random variables with variances \sigma_a, \sigma_b, and \sigma_n, respectively. Find the MAP ... 1answer 21 views ### I'm unsure of the setup for this probability question from the society of actuaries The answer is 0.223584. Here is my attempt: Company A: \mu = 10000\\ \sigma = 2000\\ \text{40% chance of at least one claim} Company B: \mu = 9000\\ \sigma = 2000\\ \text{30% chance of at ... 1answer 71 views ### Examples when vector (X,Y) is not normal 2D distribution, but X and Y are. My question is: do you know any examples when X and Y are both normally distributed, but the two dimensional vector (X,Y) is not? I found some example in book, but I don't understand it. The ... 1answer 18 views ### Linear transformations in normal distributions I am still a bit new to this topic, and was wondering if someone could check my work, it is a short exercise. Find the distribution of X = \mu + N(0,1) If we let Z \sim N(0,1) then X = \mu + ... 1answer 89 views ### find distribution of hypothesis testing? [closed] Suppose x_1,x_2,...,x_{20} is a random sample from a normal population with mean = 0 and variance  \sigma ^2 . I want to test the hypothesis H_0: \sigma ^2 \geq 4 against the alternative H_1: ... 0answers 24 views ### a question in Stat. aboout chi-square & standard normal Assume U~\chi^2(5), V~\chi^2(9), Z~N(0,1), U, V, Z are mutually independent, calculate: a. P(Z > 0.611V^\frac{1}{2}) b. P(\frac{U}{V} < 1.933) c. Find a c such that ... 0answers 38 views ### Expectation of a Rayleigh-quotient-like form for normal random vectors I have been trying to calculate or find a result for the expectation\mathbb{E} \left[ \frac{w^\top D^2 w}{1 + w^\top D w} \right] $$where$$w \sim \mathcal{N}(0,I_N), and $D \succeq 0$ is a ...
Similar to the question I asked before, with one subtle difference. If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are correlated with $\rho = 0.5$ then find: $a)$ the covariance ...