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

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Probability that arithmetic mean > 171 cm, for a normal distribution

I am wondering how to solve questions like this one: There are a group of men are of heights which are normally distributed with μ = 173 cm and σ = 20 cm. A random sample of 300 men is chosen. What ...
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998 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 ...
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66 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: ...
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2k 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? ...
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8k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
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395 views

Upper Limit in normal distribution?

(Iv already solved the a) part with the answer 0.2119, which is correct. The b) part asks for the upper limit, I dont know what an upper limit is in these type of questions. Can any one give me ...
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1k views

Why is normal distribution more accurate than binomial distribution?

I'm having a tough time understanding this. This is what I am told about comparing the two: The probability that Saredo is late for school is 0.6. What is the probability that in one month she is ...
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169 views

Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
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1answer
451 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
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112 views

Mentally Estimating the Normal CDF

More than once I have seen this sort of frustrating question on a Mathematics GRE practice test: A fair die is tossed 360 times. The probability that a six comes up on 70 or more tosses is... a) ...
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1answer
69 views

Can the the multivariate normal distribution be one dimensional?

Can the the multivariate normal distribution be one dimensional? Or should you then just use the normal distribution? I mean does an one dimensional multivariate normal even make sense?
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672 views

Rayleigh distribution

I have this question from my statistical theory course: A sniper shoots at a target. X and Y measure its deviation on the x and y axes. X and Y are independent and are distibuted normally with mean=0 ...
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196 views

what is the covariance matrix for deterministic signal+normal noise

Say that we have a signal that is written as follow $y=y_0+r$ where $y$ and $y_0$ are n-dimensional vectors and $r$ is n-dimensional noise vector. I would like to have $r\sim \mathcal{N}(0,\Sigma)$ ...
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98 views

About the differential entropies of well-known continuous distributions

Assume that the continuous random variable $X$ has a distribution (in a closed form expression) with differential entropy $h(X)$. Q) Then, is it true for any continuous distribution that the ...
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1answer
69 views

Probability involving dependent normal variables

I have two independent, identically distributed normal random variables $X \sim N(0,\sigma^2)$, $Y \sim N(0,\sigma^2)$. I want to know $$Pr[X-Y>4\sigma \text{ and } X<3\sigma \text { and } ...
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444 views

Normal Distribution from Standard Deviation?

So I have a data set $(x_{1},y_{1}), (x_{2},y_{2}),\dots,(x_{n},y_{n})$ and from it I have the values of $\sum x$, $\sum x^{2}$, $\sum y$, $\sum y^{2}$, $\sum xy$. My question is, how do I find a ...
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1answer
454 views

One over a Normal Distribution

If X is a normal distribution $N(0,\sigma^2)$ is $\frac{1}{X}$ any sort of "official" distribution or something that should just be computed? In particular I'm looking to find the expectation ...
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292 views

Scaling of a multivariate normal

We know that if a variable $X$ is iid from a $N(\mu,\sigma^2)$, the distribution of $X+b$ is $N(\mu+b,\sigma^2)$ If we scale the $X$ by a scaling factor $k$, the new distribution will be ...
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74 views

Statistical Problem (part 2)

Following my question I found another problem. Having the same data from the other question: There are 2 melon stores. The melon weights follow a normal distribution. Store A -> μ = ...
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311 views

Well known probability distributions defined on a $n$-dimensional simplex besides the Dirichlet distribution?

Are there well known probability distributions defined on a n-dimensional simplex besides the Dirichlet distribution where the variation of of each component doesn't vary as much when the mean of the ...
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1answer
429 views

Density of truncated normal distribution?

I have a truncated normal distribution with mode $0$ and variance $\sigma^2$ that only consists of non negative values. What is the density of this distribution at some non negative $x$? I have just ...
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346 views

General characteristics of a normal distribution

If the normal distribution curve is symmetrical about the vertical line then the mean = mode = median This would mean that: ...
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267 views

μ and σ calculations for a random variable with a normal distribution [duplicate]

Possible Duplicate: Calculating mu and sigma (μ and σ) of a normal random variable If I have a random variable X with parameters μ and σ unknown. It is known that $P (X \ge 75) = 0.7764$ ...
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1k views

Approximating a sum of exponential distribution with a normal distribution

Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the ...
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1answer
516 views

How to directly compute an integral which corresponds to the normal distribution

How does one directly (by finding primitive) compute an integral which corresponds to the normal distribution: $$\int_{a}^{b} e^{{-(x-a)^2}/{2s^2}} \,\mathrm{d}x$$
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127 views

Calculating the “jaggedness” of a distribution

I'm sure "jaggedness" isn't the right term to use here, so please correct me. I'm trying to quantify how jagged a distribution is. For example, this is moderately jagged: distribution #1 This is ...
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25 views

Confidence Intervals

Let $x_1, \ldots, x_n$ be a sample from a normal population having unknown mean and variance. Let $\bar{x}$ be the average of the first n of them. What is the distribution of $x_{n+1} - \bar{x}$? If ...
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64 views

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}$dx?

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $$\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}dx?$$ I just need to solve this integral. But, I don't know how. ...
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49 views

Let $X$ and $Y$ be of the same dimension and jointly normal. Find the distribution of $X+Y$.

Let $X$ and $Y$ be of the same dimension and jointly normal. Find the distribution of $X+Y$. Can we start off by saying that if $X$ and $Y$ are jointly normal, then $X$ and $Y$ are normal as well?, ...
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1answer
44 views

Mean of exponential Brownian motion

I am new to stochastics and I am trying to compute the expectation of $S_t = e^{\sigma W_t}$, where $W_t$ is a standard Brownian motion and $\sigma>0$. My attempt (using the log-normal PDF here and ...
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1answer
21 views

Finding the probability of X_Bar with sample variance included?

The question I am asked is $P(\bar{X} > 3 + 0.4984S)$, where I am additionally provided $n = 25, \mu = 3.0, \sigma^2_\text{pop} = 3.0$. $\bar{X}$ is the sample mean and $S$ is the sample variance. ...
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39 views

Find the $E[X^3]$ of the normal distribution

Find the $E[X^3]$ of the normal distribution with mean μ and variance $σ^2$ (in terms of $μ$ and $σ$). So far, I have that it is the integral of $x^3$ multiplied with the pdf of the normal ...
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1answer
41 views

Expectation of normal distributions

Let $X$ have a normal distribution with mean $µ$ and variance $σ^2$. Find $E[X^3]$ (in terms of $µ$ and $σ^2$). the pdf of this function is $$\frac{1}{σ\sqrt{2\pi}} e^{\frac{-(x-µ)^2}{2σ^2}}$$ ...
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1answer
17 views

Yearly demand distribution with monthly demand given

Currently working on this problem. Monthly demand of a product has been observed to follow a normal distribution with mean of $50$ pieces and standard deviation of $5$ pieces. Assume each month is ...
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1answer
45 views

Expected value of a maximum of two draws compared to expected value of each

I am no mathematician, so I apologise in advance for not explaining myself properly, and for asking something that is probably utterly obvious for most of you. The question has to do with the ...
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1answer
14 views

Using normal distribution to create confidence interval

Let $Y~N(\mu, 1)$. Use the fact that $P(\left | Y-\mu \right | < 1.96\sigma) \approx.95$ to construct an interval $(a(Y), b(Y))$, such that the probability $\mu$ is in the interval is approximately ...
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146 views

Sum of Two Independent Normal Random Variables (Weird Z?)

So by Property $X + Y \sim \mathcal{N}(1+1.5, 0.1^2 + 0.3^2) = \mathcal{N}(2.5,0.1)$ $$P(1<X+Y<1.3) = P\left(\frac {1.0-2.5}{(0.1)^.5}<Z<\frac {1.3-2.5}{(0.1)^.5}\right)$$ Which is equal ...
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22 views

finding density of $1/Z$ when $Z$ is a standard random normal variable

If $Z$ is a standard normal r.v., we know that its density is $$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2},$$ where $-\infty \leq z\leq \infty$. I want to find what $f_{1/Z}(z)$ is. I let $Y=1/Z$, so ...
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25 views

find the value a such that the probability that a tire lasts more than a miles is approximately 0.8

The lifetime of a certain tire has normal distribution with $\mu = 50,000$, and $\sigma = 5,000$. (a) Find the probability that the tire will last between 48,000 and 56,000 miles. Let $L$ = ...
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1answer
18 views

When the covariance matrix $\Sigma$ of the p.d.f. of a m.n.d. is a $1 \times 1$ matrix (a scalar)

I was looking at the Wikipedia article talking about the multivariate normal distribution and specifically I was looking at the section talking about the probability density function of that ...
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1answer
16 views

Finding conditional distribution in multinormal case

[S]uppose that $X_1$ (sales), $X_2$ (price), $X_3$ (advertisement), and $X_4$ (sales assistants) are normally distributed with: $$ \mu = \begin{pmatrix} 172.7 \\ 104.6 \\ 104.0 \\ 93.8 ...
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84 views

Estimate the mode of the binomial distribution without Stirling's formula

Context: Let $\tilde B_n$ be standardized binomial distributed with $p\in(0,1)$ be the probability of success in the $n$ Binomial trials. So $P(\tilde B_n = x_k)=\binom nk p^k q^{n-k}$ for $x_k = ...
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1answer
17 views

Meaning of numbers in covariance matrix

For a course I'm following we need to work with multivariate guassians. In this case there are four variables $x_1$ through $x_4$ with the specified covariance matrix. Even though the matrix is ...
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1answer
16 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = ...
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39 views

Normal distribution with P(x=a) and P(x≥a)

Given: height of $1000$ students normally distributed with $\mu=174.5\,\mathrm{cm}$, $\sigma=6.9\,\mathrm{cm}$ Find: a. $P(x<160\,\mathrm{cm})$ b. $P(x=175\,\mathrm{cm})$ c. ...
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49 views

Is there a simpler way to calculate correlation?

Let's consider that a variable y constructed from x $x_i ∈ \left\{1,3,5,7,8\right\}$ $f(x_i)=2x_i+1$ $y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $ where $ε_i$ is a ...
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1answer
22 views

Choosing the variance for a Gaussian Distribution for the Langevin Equation

I am trying to numerically integrate the Langevin equation which is given by $$ \ddot x= -\frac{6\pi\eta a}{m} \dot x + \frac{\xi(t)}{m} $$ where $\xi(t)$ is a probability distribution. I'm ...
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56 views

Lognormal distribution function

What exactly is the Lognormal distribution? Also how can I find it's distribution. I came across the following problem in Sheldon M Ross, I am not understanding where to start. Please help A random ...
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85 views

Bivariate normal distribution of points

I would like to generate points (x,y) in a 2-D plane that has a circular normal distribution similar to this: I found multiple terms for describing a "circular normal distribution" and yet, I'm not ...
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27 views

Random variable with density function that is scaled geometric mean of density functions of two independent normally distributed random variables

Given two independent normally distributed random variables A and B: $$A \sim \mathcal{N(\mu_A, \Sigma_A)}$$$$B \sim \mathcal{N(\mu_B, \Sigma_B)}$$ is there a way to find normally distributed random ...