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

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2
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1answer
659 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
110 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
112 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
478 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: ...
1
vote
1answer
350 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 ...
2
votes
2answers
294 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
vote
1answer
70 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
199 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
39 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
151 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
66 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
230 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
4k 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
306 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
31 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
751 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
299 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
68 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
98 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
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0answers
124 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
129 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
118 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
124 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
89 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
129 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
50 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
1answer
580 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
191 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
37 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
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1answer
142 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
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1answer
95 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
29 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
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2answers
75 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
168 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
141 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
149 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
42 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. ...
0
votes
1answer
193 views

Mathematical Statistics (Normal Distribution)

The weights of a population of mice fed a certain diet follow a normal distribution with mean μ=100 grams and standard deviation σ=20 grams. A random sample of 8 such mice is taken. (a) Find the ...
1
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1answer
276 views

Tail inequalities for multivariate normal distribution

There exists an closed expression for univariate normal CDF, together with simpler upper-bounds under the form, $$ \Pr\big[X > c\big] \leq \frac{1}{2}\exp\Big(\frac{-c^2}{2}\Big)~, $$ $$\text{where ...
1
vote
0answers
36 views

Finding the distribution under a new measure

Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where $Z_t$ are standard normal variables. Find the distribution of ln($S_1/S_0$) under the Q measure given that dQ/dP is ...
0
votes
1answer
69 views

Expected highest sample from N samples of a normal distribution?

Given a normal distribution, how would I determine what the expected highest sample would be out of N samples? Presently I'm doing some strange calculations that I think are incorrect; I'm solving ...
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0answers
34 views

Confusion related to gaussian

I have this confusion related to gaussian distribution Do we need to have something like $e^{-\frac{x^2}{2}}$ to be called gaussian or $e^{-{x^2}}$ is enough to be called Gaussian. I was reading this ...
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2answers
38 views

Normal Distribution - places to look for it

If i were to look around for items, objects or any 'samples' for that matter, which ones would give me a normal distribution? I know heights and weights of people could give me a normal distribution. ...
0
votes
1answer
64 views

Confusion related to gaussian distribution

I was reading this paper where it had a gaussian distribution model. I mean gaussian is given by $P(y) = \frac{e^{-\frac{1}{2}(y -\mu)^T \Sigma^{-1}(y -\mu)}}{2\pi^{n/2}|\Sigma|^{1/2}}$ But is ...
0
votes
1answer
723 views

Normal approximation to the log-normal distribution

Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds ...
1
vote
1answer
184 views

What proportion are above x of sample size n where X ~ N(0,1) Homework

I have a homework question that I'm not quiet sure of. It follows as so: Consider a random variable $X$ that has a standard normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. ...
5
votes
2answers
189 views

expectation equations

I am just trying to understand the following three equations. $\phi(x)$ denotes the standard Gaussian cumulative distribution function and $X$~$N(\mu,\sigma^2)$ (1) $\mathbb{E}[e^{tX}f(X)]=e^{\mu ...
0
votes
1answer
162 views

confidence interval of binomial disribution using standard deviation

Just as the normal distribution has the 68–95–99.7 rule with 68% of the data within +- 1 standard deviation and so on, does the binomial distribution too has something like that. Or does its being a ...
0
votes
1answer
265 views

Prove that the Q function is bounded such that $Q(x)\le\frac{1}{2x^2}$

Prove that the gaussian Q function is bounded on the top by $\frac{1}{2x^2 }$, i.e. $Q(x)\le\frac{1}{2x^2}$ for $x\ge0$, using the chebyshev inequality and the nakagami-m distribution with m=0.5(that ...
0
votes
1answer
189 views

Is there a name for the normal CDF function $\Phi(\cdot)$?

I can't seem to find a plain English name for the CDF of the normal distribution $\Phi(x)$. However, I am aware of several other related functions that have a name, so I feel like this one should as ...