# Tagged Questions

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### normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
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### Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
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### Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
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### convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
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### Vector distribution after Girsanov transform

Let $X$ be a gaussian vector under $P$ and $U$ a variable such that the vector $(X,U)$ is gaussian. $dQ = Y dP$ with $Y = e^{(U −E_p(U) − 1/2 var_p[U])}$. I have to show that $X$ is gaussian under ...
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### Determining $\sigma$ given mean and proportion of a Normal distribution?

The marks of a random sample of students with mean $\mu$ and standard deviation $\sigma$ showed that 15.87% scored higher than 70. The distribution of the marks is Normal with mean $50$ standard ...
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### An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
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### Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
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### Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
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### Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
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### Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
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### What is the reason for the one-half in the normal pdf's gaussian (i.e. : why $\exp(-x^{2}/2)$ instead of $\exp(-x^{2})$ )

It doesn't seem to relate to normalization, as the normalizing constant adapts to every possible "upstairs formulation", and in the standard case is $\displaystyle\frac{1}{\sqrt{2\pi}}$. Does it ...
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### Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
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### Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
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### Binomial distribution vs Normal distribution

It is often said that the normal distribution "approximates" the binomial distribution. What is the precise mathematical expression of this fact?
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### Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
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### Expectation of normal CDF with truncation

Suppose that $a$ and $T$ are given positive numbers. I would like to evaluate \begin{align*} \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right],\tag{1} \end{align*} where ...
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### Modeling Gaussian Error

Context I am designing a simulation of a robot receiving input from a sensor which has gaussian error. The robot will start from a known position and move at a constant speed; the sensor will ...
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### When do normal distributions not occur?

I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. For example, suppose ...
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### Calculation of distribution of a gaussian process

Currently finishing the last year of PhD in statistics, we wonder if you could help us with the following. Let $T = [0,1]$ and $X = \left( X_{t}, t \in T \right)$ be a gaussian process with mean ...
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### What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
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I have independent variables $X_1, X_2,\ldots,X_n$ with normal distribution on range $[0,1]$ . Next, variables $Z_i$ are created according to this formula $Z_i = - \frac{1}{\lambda} \ln(1-X_i) ... 1answer 42 views ### Normalizing a dataset from the interval [0,1] with fixed properties. So I have a rather large dataset where values are from the interval$[0,1] \in \mathbb{R}$. But the problem is that a big portion of the values are extremely close to$0$. So firstly I'm looking for ... 2answers 14 views ### Determining the marginal distribution Consider$X=(X_1,\ldots,X_n)^T\sim\mathcal{N}(\mu,V)$. Show that then$X_i\sim\mathcal{N}(\mu_i,V_{ii})$for all$1\leqslant i\leqslant n$. Good day! Ok, I have to determine the marginal ... 1answer 31 views ### Probability distritubion of linear function Given a variable X belongs to gaussian distribution$N(\mu, \sigma)$. How to find the distribution of linear function$y=ax+b$? My answer is that the linear distribtion will belong the ... 1answer 15 views ### Normal Distribution how$N(x-x_n|0,\sigma^2) = N(x |x_n,\sigma^2) $I read an expression Could someone explain the step$N(t-t_n|0,\sigma^2) = N(t | t_n,\sigma^2) $? 1answer 43 views ### Standard deviation with multiple means and deviations The amounts of a certain mineral that can be produced in a day from mines$1$,$2$, and$3$are independent normal random variables with means equal to$80$,$90$, and$75$pounds, respectively, ... 1answer 23 views ### Exponential deviation with two$x\$ values

I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ...