Tagged Questions

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

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0
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1answer
161 views

Probability with bullets and walls

There are two shooters with different guns and bullets. Each shooter shoots a bullet to a different target hanging on a wall. The hit of each bullet follows a normal distribution centered on its ...
2
votes
2answers
103 views
0
votes
1answer
31 views

normal as approximation to binomial

Among 784 checks, 479 had amounts with leading digits of 5, but checks issued in the normal course of honest transactions were expected to have 7.9% of the checks with amounts having leading digits of ...
2
votes
1answer
160 views

determine Fisher information of $N(0,\sigma^{2})$ over $\sigma^{2}$

So far i've got that $I(\sigma^{2}) = E_{\sigma^{2}}[\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(X)]^{2}$. And i got that $\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(x) = ...
2
votes
0answers
39 views

Mixture of Gaussians — Distribution Weight

I've been having trouble understanding how to simplify (as well as understand) the equation for what I'm calling the "Distribution Weight" of a Conditional Mixture of Gaussians distribution. Namely, ...
1
vote
2answers
92 views

How big of a sample size is necessary to be sufficiently confident in predictions?

A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be $90\%$ confident that her estimate is within $2$ ...
0
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1answer
39 views

Multiple independent random number streams

This question is somehow related to this one. Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution I want to do Monte Carlo simulations in ...
0
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0answers
44 views

Conditionally normal distribution with a normal mean

The following question is part of my homework: Suppose that $\mu \sim N(0, 1/\alpha)$ and $x|μ \sim N(\mu, 1/\beta)$. By integrating out $\mu$, show that the marginal distribution of $x$ is given by ...
0
votes
2answers
91 views

Theoretical impossibility? Deviation from normality with a sample greater than 300?

Huge thanks in advance! I've been lead to believe that the following is a theoretical impossibility: a population larger than 300 records without an approximation of a normal distribution. The ...
1
vote
1answer
74 views

Generating 2D random vector from 4D covariance matrix

I have such covariance matrix $C$: ...
0
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0answers
65 views

Asymptotic (in)dependence of a maximum of an i.i.d. sequence of Gaussian random variables on a single random variable in this sequence

Suppose that I have a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$ where $X_i\sim\mathcal{N}(0,1)$. Denote the maximum of this sequence by $M_n=\max(X_1,\ldots,X_n)$. I ...
0
votes
1answer
91 views

Finding the variance of a normally-distributed random variable

X is a normally-distributed random variable, and $P[X<20] = 1/10 = P[X>100]$ I am trying to solve for the mean and the variance. I know that $\mu=60$ by symmetry. How can I solve for ...
1
vote
1answer
429 views

MLE of fourth moment of normal distribution

Take $X\sim N(0,\theta)$, and let $\phi = E(X^4)$, the fourth moment. What is its MLE, $\hat{\phi}$, and what is the asymptotic distribution of $\sqrt{n}(\hat{\phi} - \phi) $ as $n\to \infty$? Any ...
1
vote
1answer
103 views

Finding the expectation of functions of random variables with a bivariate normal distribution

X and Y have a bivariate normal distribution. I am given that $E[X] = 4$ and $E[Y] = 10$. I am asked to find $E[X^2 - Y^2]$ WITHOUT integration. I know how to solve for this using integration, but ...
1
vote
3answers
76 views

$Z$ score probability

I was given a question where I was supposed to find the probability of obtaining $y$ between two scores, however when I input my answer it tells me that I'm wrong, the question is given below along ...
5
votes
2answers
320 views

Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables

It's well known that, for a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$, where $X_\max=\max(X_1,\ldots,X_n)$, the following convergence result holds: ...
2
votes
1answer
244 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
1
vote
0answers
153 views

Fourier transform of a complex exponential with quadratic argument

I'm a PhD student who is starting to work right now in the well-established field of ultra-fast optics. The thing is that, in most of the papers I have been reading during the past few days, there is ...
1
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1answer
66 views

Bigger than and equals rewritten in normal distribution question

So it is correct to say that $P(482\le x \le 510) = P(x \le 510) - P(x < 482)$ where x is a random variable in a normal distribution? Thanks!
1
vote
2answers
134 views

Calculating integral with standard normal distribution.

I have a problem to solving this, Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate. However, at the bottom I ...
1
vote
1answer
36 views

Probability , Geometric and Gaussian

So,I'm good at the questions which require the understanding of basic formulaes , but this one my prof said needs me to think (for the first one)'geometrically'=Stumped. Please Help! The second is an ...
1
vote
0answers
127 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
0
votes
1answer
61 views

normal distribution expected value

In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html $$f(x) = \sqrt{\frac{k}{2\pi}}e^{-\frac{k(x-\mu)^2}{2}}$$ they let $x-\mu = v$ $dx = dv$ and conclude that ...
3
votes
2answers
189 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
0
votes
1answer
3k views

How to calculate the probability of a normal distribution with unknown mean and unknown variance?

How do you calculate the probability of a normal distribution with unknown mean and unknown variance? If a problem stated, for example, that 15% of the time sales are more than 15,000 and 20% of the ...
1
vote
1answer
127 views

Calculating an average on normal distribution

Given the fair dice, if the result is $1$ or $2$ the profit is $3$USD, if the result is $6$ you don't win or lose anything, for every other result you lose $2$USD. What is the average profit, that ...
0
votes
1answer
27 views

In statistics, what is the meaning of $Z_{0.3}$

What is the meaning of $Z_{0.3}$ and how do I calculate it? I know it was calculated this way: $$Z_{0.3} = -Z_{0.7} = -0.52$$ I tried to follow the General Distribution table but I can't seem to ...
1
vote
1answer
466 views

Normal distribution, chi-square distribution and t distribution combiened

How to prove that when X is from Normal Distribution and Y is from Chi-square Distribution with parameter f and X,Y are independent then X/sqrt(Y/f) is from t distribution with parameter t? I got ...
2
votes
2answers
200 views

Conditional mean and variance of normal random variables

There are two independent normal random variables $N_1, N_2$ with means $\mu_1, \mu_2$ and variances $\sigma_1^2, \sigma_2^2$ respectively. Is there a way to compute the two conditional expressions ...
0
votes
1answer
66 views

What is the effect of the variance on a sequence of cumulative product?

We randomly draw numbers from a normal distribution with mean equals $mu$ and variance equals $var$. We draw the values: $x_1, x_2, x_3, x_4, ...$ Then, we construct a sequence made of the ...
1
vote
1answer
81 views

Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
0
votes
0answers
99 views

Summing many non-standard i.i.d. uniform random variables

all! I have looked up a fair bit on this question and learned much about the problem. But haven't been able to get any crisp answers. Sorry, if I'm missing something obvious. I know one can use the ...
0
votes
2answers
111 views

Multivariate normal distribution from invertable covariance matrix

I want to generate a random vector with $\mathcal{N}(0, C)$ distribution, i.e. normal distribution with $0$ mean and given covariance matrix $C$. $C$ is not invertible (singular). Here it's written: ...
1
vote
1answer
82 views

The MLE of a $N(\theta, 1)$ distribution

I am trying to find the Maximum Likelihood Estimator of an i.i.d. sample $X_1, \ldots, X_n$ arising from the model $N(\theta, 1)$, where $\theta \in [0,\infty)$. I have done this problem previously ...
3
votes
0answers
100 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
1
vote
2answers
88 views

Bound of Standard Normal Integral

Consider the Standard Normal Integral given by: $$ I=\int_{-\infty}^{\infty} \frac{1} { \sqrt{2\pi} } e^{ \left( -z^2 /2 \right)} dz $$ In order to prove that it exists we note that the integrand is ...
2
votes
1answer
111 views

The probability that a randomly chosen grain weighs less than the mean grain weight

If Y has a log-normal distribution with parameters $\mu$ and $\sigma^2$, it can be shown that $E(Y)=e^\frac{\mu + \sigma^2}{2}$ and $V(Y)=e^{2\mu +\sigma^2}(e^{\sigma^2}-1)$. The grains composing ...
0
votes
2answers
44 views

Why are vectors $X_2$ and $X_3$ bivariate normally distributed?

I have a stochastic vector $\mu = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$ and $\Sigma= \begin{bmatrix}1 & 0 & -1\\0 & 2 & 0 \\ -1 & 0 & 3\end{bmatrix}$. I have to proof that ...
0
votes
0answers
67 views

Product of a univariate and multivariate gaussian

If I have a multivariate gaussian distribution as P(x) = $N(\mu,\Sigma)$ and another univariate distribution given by P(y) = $N(\mu1, \sigma)$, what will be p(x,y). It will be gaussian but what will ...
0
votes
1answer
413 views

Exponential and Uniform distribution with conditional probability

A computer lab has two printers. Printer I handles 40% of all the jobs. its printing time is Exponential with the mean of 2 minutes. Printer II handles the remaining 60% of jobs. Its printing time is ...
0
votes
1answer
154 views

Forumla for finding conditional variance

I need to find the conditional variance $\mathop{\mathrm{Var}}(X_1|(X_2+X_3))$, given that $X_1\sim N(0,1)$ and $X_2+X_3\sim N(0,2+2\gamma)$. The covariance between X1, X2+X3 is $\rho$. From this ...
2
votes
1answer
35 views

Calculating probability of difference of two distributions.

A has normal distribution of scores of students with $X \sim \mathcal N(625, 100)$ B has normal distribution of scores of students with $X \sim \mathcal N(600, 150)$ Now I have to calculate ...
0
votes
1answer
115 views

Joint Probability Distribution of a Gaussian Random Variable Correlated with a Gamma Random Variable

I want to know if the joint PDF of a Gaussian RV correlated with a Gamma RV can be found in closed form. The correlation is known.
0
votes
1answer
83 views

$\sum(y_i-\bar{y})^2$ can be written in the form $\sigma^2 X'AX$ where $X\sim N(0,1)$. What is $A$?

Random sample $Y_1,\dots, Y_n$ of size n from a univariate normal population with ($\mu, \sigma^2$). Let $\bar{y}=\frac{1}{n}\sum Y_i$. $\sum(y_i-\bar{y})^2$ can be written in the for $\sigma^2 X'AX$ ...
0
votes
2answers
24 views

Finding mean given information

Given that 95% of the values is between 20 and 34, what would be the mean? I think it's 27..but I'm not sure..if it's not 27, what's the right way to solve it? Please explain this to me, thank you.
0
votes
2answers
58 views

Normal distribution

I have this question: A normal distribution is such that 16% of it is smaller than 13, and 2.5% of it is larger than 22. What's the mean of this normal distribution? I know I should be using the ...
0
votes
0answers
106 views

unbiased estimator of the area of the circle

the radius of a circle is measured with an error of measurement which is distributed normal with mean $0$ and variance $\sigma^2$,$\sigma^2$ unknown.Given $n$ independent measurements of the radius , ...
0
votes
1answer
113 views

Conditional Normal Distribution of Mice

The weights of a population of mice fed a certain diet follow a normal distribution with mean $\mu=100$ grams and standard deviation $\sigma=20$ grams. A random sample of $8$ such mice is taken. Let ...
0
votes
1answer
117 views

Joint distribution of two marginal normal random variables

Question: Suppose we have: \begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ 3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 2 \end{bmatrix} \right) ...
2
votes
1answer
35 views

What distribution do the rows of the Stirling numbers of the second kind approach?

In wikipedia about the Pascal triangle: Relation to binomial distribution "When divided by 2n, the nth row of Pascal's triangle becomes the binomial distribution in the symmetric case where p = 1/2. ...