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

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3
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2answers
199 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
167 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
28 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
564 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
266 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
67 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
87 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
106 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
113 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
94 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
106 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
114 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
45 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
461 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
173 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
36 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
124 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
87 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
60 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
134 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
127 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
122 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
38 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
165 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 ...
0
votes
1answer
196 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
32 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
66 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 ...
1
vote
0answers
33 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 ...
0
votes
2answers
37 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
63 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
527 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
146 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
184 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
130 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
140 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 ...
1
vote
0answers
236 views

Almost sure convergence of maximum in a sequence of Gaussian random variables

Let $X_1, X_2,\ldots,X_n$ be an i.i.d. sequence of standard Gaussian variables and $M_n=\max(X_1, X_2,\ldots,X_n)$. I am trying to understand the mechanics of the proof of almost sure convergence ...
0
votes
3answers
146 views

conditioned expectation of X wrt the product XY

I have to calculate $\mathbb{E}(X|X*Y)$ with X,Y being independent and standard normal distributed. I got at tip in this post (Conditional expectation on components of gaussian vector), that I should ...
0
votes
1answer
34 views

Recognizing that a function is a standard normal distribution with a certain mean and variance

In an example, I am told that $$\int_0^T{e^{-bt}}\,dt=\frac{1}{b}(1-e^{-bT})$$ is $n(0,T)$, i.e., which I took to mean normally distributed with mean of $0$ and variance of $T$. Note: The value of ...
0
votes
1answer
381 views

Conditional expectation on components of gaussian vector

I think I got the definition of the conditional expectation now, but I'm still having some problems with actual calculations... Let $(X,Y,Z)$ be a real gaussian vector. X and Y centered and ...
0
votes
2answers
31 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
248 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
230 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 ...
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
311 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
99 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 ...