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

learn more… | top users | synonyms

0
votes
2answers
30 views

Error in solving for raw score; incorrect formula used?

According to a study of how long a person is willing to wait for their flight, it is found that the mean time a person is willing to wait is 5.2 hours with a standard deviation of 1.1 hours. Consider ...
0
votes
1answer
232 views

Probability of a normal distribution; more than, less than confusion.

You are interested in finding how many hours a person is willing to wait for a plane. It is found that the time people are willing to wait has a $μ = 5.2$ and a $σ = 1.1$. What is the probability a ...
1
vote
1answer
183 views

Given a normal distribution, how do you determine a proportion that is outside of a range?

I am presented with the question: The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a ...
1
vote
0answers
24 views

Normal Distribution and test hypothesis

I have done 3 experiments. For each one of them, I have repeated the same experiment 100 times. Which gives me three sets of 100 numbers. Experiment 1: for number 30 ---> 100 results Experiment 2: for ...
0
votes
0answers
84 views

Normal distribution probability calculations

The following normal distributions represent race finishing times for a group of swimmers (S1 to S6). For example, Swimmer 1 (S1) has a mean finishing time of 60 seconds with a standard deviation of ...
4
votes
1answer
1k views

Expected values for normal distribution

So I have a practice question on an example exam, and I am a bit stumped by it: Suppose that $X \sim N(1,2)$. Find: $$ E((X−1)^4) $$ and $$ E(X^4) $$ I am a bit confused as to how to proceed. Now, ...
1
vote
1answer
283 views

Normal Distribution Quantiles and Value at Risk

I'm preparing an exam, Quantitative Methods for Financial Markets. My book is not really clear for what concerns the calculation of normal distribution quantiles that have to be used in VaR's formula. ...
1
vote
2answers
98 views

Standard Normal Distribution to Normal Distribution

Given function f() which returns a random value from a standard normal distribution, is it possible to define a function g(mu, sigma) which returns a random value from a normal distribution with a ...
0
votes
1answer
38 views

Normal distribution with less Probability [closed]

How do I calculate with Excel's formula to answer the question: What is the probability for a student's point is less than 560 by using excel? I know how to ...
-2
votes
1answer
47 views

Probability and expectation involving four standard normal RVs [closed]

Suppose that $Z_1, Z_2, Z_3$ and $Z_4$ are independent standard normal random variables. Find: (a) $P(Z_1 + Z_2 > Z_3+Z_4+1)$ (use the table of standard normal probabilities) (b) $E(4Z_1 + 3Z_2 - ...
0
votes
0answers
83 views

Limiting distribution of standard Negative Binomial Random Variate.

Let $X_{\alpha}\sim NB(\alpha,p)$ $$P(X=k)=\binom{\alpha+k-1}{k}p^k(1-p)^{\alpha};\quad x=0,1,\ldots$$ $\alpha>0$ and $0<p<1$ let $$W_{\alpha}=\frac{X_{\alpha}-\mathbb ...
2
votes
0answers
34 views

Combining two circulating normal distributions

I am working in estimating the impact of location error on location based services. My question is listed below. If the error distribution of location estimation follows in general a normal ...
0
votes
1answer
97 views

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need?

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need? I may not be using the correct terminology so here's a graph: Based on this, if you ...
0
votes
1answer
202 views

Standard deviation of less than one

How would I find the approximate percentage of values within a standard deviation of less than one on the normal model? Chebyshev's rule is only used when the standard deviation is greater than or ...
1
vote
1answer
171 views

normal distribution derivation

In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html how do these equal? $$ -k\int (x-\mu) dx = -\frac{k}{2} (x-\mu)^2$$ Isn't this the case? $$ -k\int (x-\mu) dx ...
-1
votes
2answers
93 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
1
vote
2answers
56 views

$X$ is half normal and $S ∼ U{(−1, +1)}$. How $Z = SX ∼ N(0, 1)$?

If we chop a standard normal distribution in half and use only the positive side (scaled up by a factor of $2$ to maintain a proper density), then we get the so-called ‘half normal’ density: ...
3
votes
2answers
411 views

How to approximate the integral of the standard normal distribution.

So I have this eqn. $$ f(x)= \frac {e^ \frac{-x^2}{2}} {\sqrt{2\pi}} $$ I need to find: $$ \int\limits_{-1}^1 f(x)dx $$ So I want to use this series to integrate. I know that: $$ e^x = ...
0
votes
2answers
548 views

How do you determine the sample size of a normal distribution?

I am presented with the question: The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is ...
0
votes
1answer
64 views

Approximating the optimal value of a function involving a Gaussian integral

Consider the following function $$ f(\lambda) = \alpha (1+\lambda^2) + (1-\alpha)2\int_\lambda^\infty (x-\lambda)^2 \phi(x) dx $$ where $\alpha \in (0,1)$ and $\phi$ is the standard normal probability ...
0
votes
0answers
25 views

Determining Range

$X_i\sim^{iid}N(0,1);\quad i=1,2$ so $x_i$ ranges from $-\infty$ to $\infty$. Now $Y=X_1^2+X_2^2$ so $y$ ranges from $0$ to $\infty$. But how $Z=X_2$ is ranges from ($-\sqrt y$) to $\sqrt y$ ?
0
votes
1answer
48 views

Suitability of skew normal for rating task and calculation

in an experiment, I ask participants to rate qualities on a continuous scale. I expect the results to be normal distributed and I am confident that assuming a normal works fairly well for most values. ...
1
vote
0answers
46 views

How to learn mixture Gaussian with inequality constraint of component variances

Let $f_1(x)$,…,$f_n(x)$ be Gaussian density functions with different parameters, $\mu_i$ and $\sigma_i$ are the parameters (mean and variance) of the Gaussian component i, and $w_1,\ldots,w_n$ be real ...
2
votes
1answer
84 views

Given a covarince matrix, generate a Gaussian random variable

Given a $M \times  M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$. Not really sure what to do here. You can calculate ...
0
votes
2answers
615 views

How do you compute this normal distribution?

The question is: Given that X is normally distributed with mean 100 and standard deviation 9, compute the following for n = 16. (a) Mean (Round your answer to the nearest integer.) and ...
1
vote
1answer
160 views

convolve probit function with gaussian [duplicate]

I want to prove the following, however, not sure where to start. $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Where $\Phi(\cdot)$ is the probit function, ...
1
vote
1answer
1k views

The difference between unbiased/biased estimator variance.

The biased MLE of Normal distribution is: $\hat{\sigma }_{MLE} = \frac{1}{N}\sum_{N}^{i=1}\left({x}_{i} - \hat{\mu }\right)^{2}$ And unbiased is: $\hat{\sigma }_{unbiased} = ...
1
vote
1answer
1k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
0
votes
1answer
310 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
3
votes
2answers
409 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
-1
votes
1answer
95 views

quotient Groups of different normal subgroups. [closed]

Let G have two normal sub groups N and M,|N|=|M|(so that |G/N|=|G/M|).Now consider their quotient groups G/N and G/M.Is it possible that for each g*n(g belongs to G and n belongs to N),there exists ...
0
votes
2answers
114 views

Normal distribution probability problem.

There are lots of salmon in a pond and their length (in centimeters) obeys normal distribution $N(70, 5.4^2)$. You and your friend go fishing and decide to continue fishing until both of you catch at ...
3
votes
1answer
94 views

Z scoring and normal distribution

I tackled a question that asked Given the mean and std dev for rainfall in a city, ...
3
votes
1answer
45 views

Independent representation of correlated $N(0,1)$ variables

Assume that $X_1$ and $X_2$ are correlated $N(0,1)$ variables. Now we can write \begin{align*} (X_1,X_2)^{T}=(\tilde{X_1},\gamma \tilde{X_1}+\sqrt{1-\gamma^2}\tilde{X_2})^{T} \end{align*}where ...
1
vote
0answers
53 views

$\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$

Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
0
votes
1answer
186 views

Ratio of dependent chi squared random variables

Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. ...
2
votes
2answers
885 views

joint probability of two Gaussian

I was studying factor analysis model using a lecture note by Prof. Andrew Ng (http://cs229.stanford.edu/notes/cs229-notes9.pdf). It says $z \sim N(0,I) \\ \epsilon \sim N(0, \psi) \\ x = \mu + ...
1
vote
0answers
94 views

Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
1
vote
1answer
47 views

Proving MLE for normal distribution

I need to prove that using maximum likelihood estimation on both parameters of normal distribution indeed maximises likelihood function. So, the log-likelihood function for parameters $\sigma$ and ...
3
votes
2answers
717 views

Sums of Products of Two Normal Variables

Suppose that $X_1 ,\ldots,X_n,Y_1,\ldots,Y_n$ are all independent normal random variables with different means and variances. What is the PDF of the following random variable? ...
0
votes
1answer
108 views

integrate moments normal distribution between finite limits

Can somebody help me to evaluate the following integral: $$\frac{1}{\sqrt{2\pi}\sigma}\int_a^b x^2 \exp\left(\frac{-x^2}{2\sigma^2}\right)\mathrm dx$$ Answer involving cumulative normal (erf) would ...
1
vote
1answer
108 views

Correlation of sums of correlated variables

I'm trying to work out an expression for a correlation of the weighted sums of two r.v.'s with a third r.v. To be precise, I have a trivariate normal distribution: $$\{X,Y,Z\}\approx ...
1
vote
1answer
971 views

How do I know if a sufficient statistic is also complete?

For example, for an i.i.d. sample of random variables $X_i$ distributed according to a normal distribution, I found a sufficient statistic—the sample mean. How do I know if this is also complete? ...
2
votes
1answer
114 views

Gaussian function

I want to scale the Gaussian function $\exp(-x^2)$ to the unit disc. In particular, I wish to represent $\int_0^\infty \exp(-x^2) dx$ as $\int_0^1 g(x) dx$, where $g$ should be the rescaled Gaussian ...
0
votes
1answer
51 views

probability in normal density function

Q: let X be a continuous random variable with NORMAL DENSITY $$f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{−(x−\mu)^2/ 2\sigma^2}$$ We know that $\mu = 70$ and $\sigma = 2$. Find $P(68 \leq X ...
7
votes
3answers
2k views

Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ...
1
vote
1answer
42 views

Question on sum of normal variable

I have a small doubt. If X and Y are standard normal variables, is $ Z=(X+Y)/\sqrt { 2 } $ a standard normal variable ? If I am correct, $X+Y$ follows $N(0, 2)$. So, Z must follow $N(0, 2 / \sqrt { ...
2
votes
2answers
947 views

Expected value for maximum of n normal random variable

Let $X_1...X_n\sim N(\mu,\sigma)$ be normal random variables. Find the expected value of $\max_i(X_i)$ and $\min_i(X_i)$. The sad truth is I don't have any good idea how to start and I'll be ...
0
votes
1answer
144 views

what is the pattern in the distribution of divisors.

I made a table that shows the number of divisors for each number less than 500, and i think that there is a pattern, for example when there is a spike in the number of divisors the surrounding numbers ...
0
votes
1answer
53 views

Get random position on surface

I'd like to get a random position on a surface of an object, and also follow it's normals. Example, let's say I have a sphere, I can get all the face, normal and vertex positions and well as their ...