# Tagged Questions

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

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### Characteristics of a Gaussian on a logarithmic scale?

I have data modelled to be Gaussian-distributed, and then displayed on a logarithmic scale and like to know some properties of the displayed data. Effektively, I ignore that a Gaussian $X$ (with ...
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### Confidence Itervals; $Z_{\alpha}$ & $Z_{\alpha/2}$

I'm confused about what exactly $Z_{\alpha}$ is, does there exist a formula for it in terms of $\alpha$? IF so, is there also one for $Z_{\alpha/2}$?
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### get data to draw a gauss curve

I would like to know how to get some data from a normal distribution to draw its gauss curve. I have the standard deviation, the average and the x, but I don't know how to get some points to draw the ...
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### How to solve $P(X=a) = P(X=b)$

A random variable X is normally distributed with $\mu = 60$ and $\sigma$ = 3. What is the value of 2 numbers a,b so that $P(X=a) = P(X=b)$. The solution is $a = 60$ and $b = 65$. However, I do not ...
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### From normal to normal standard distribution + a bonus Q

I am failing to see where the $\sigma$ is going in the below. Given that normal distribution's pdf is: $$p(x) = \frac{1}{\sigma \sqrt{2 \pi}}\exp \left( -\frac{(x-\mu)^2}{2 \sigma^2} \right)$$ and ...
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### Ratio of CDF to PDF increasing?

Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function. Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x? Numerically it seems to ...
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### Normal approximation to Binomial probability distribution

Where did this 0.5 come from? I understand we are using Z-score but in my calculations I basically omit the 0.5 to get a probability of .9616.
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### How should I interpret this notation?

I am reading some lecture notes and I'm not sure how to interpret this: $$b_j(x)=p(x\mid s_j)=N(x;\mu,\sigma^2)$$ It is clear from the context that $N$ refers to normal distribution, but what exactly ...
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### Teasing apart an explanation of the Central Limit Theorem

I'm looking at the central limit theorem, and cannot see in the explanation given to me how the average of identical distributions results in the normal distribution. I am told to consider a sequence ...
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### PDF of $Z=\frac{X^2+Y^2}{2}$ where $X\sim N(0,1)$ and $Y\sim N(0,1)$

Say $X \sim N(0,1)$ and $Y\sim N(0,1)$ are independent random variables. So: $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}x^2}$ and $f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}y^2}$. Now I am ...
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### Conditional density, bivariate normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independent. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...
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### grading on a curve using normal distribution

suppose we have a grade list: $\text{grades}=\{2,3,5,7,8,10,9,9.75,8,0,11,10,10,3,5.25,13,14,20,18,9\};$ which mean equals to 8.75 and Standard deviation is 5.06471. we want to improve the grade ...
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### Formula to get mean and standard deviation of this multi-variable equation

$$\binom n x \times\left(\frac1r\right)^x\times\left(\frac{r-1}r\right)^{n-x}$$ If you have $n$ boxes and have a $\frac1r$ chance to fill each one, this equation returns the chance that you fill ...
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### Calculating probability of a normal distribution, not getting correct answer

I'm doing a homework assignment and having some trouble matching the correct answers from my professor. As a reference, I'm calculating these answers using R. The question is as follows: Assuming ...
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### Integral of multplication of normal pdf and Rayleigh pdf distribution

I need to calculate following definite integral $$\frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx$$ It is actually ...
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### Finding expected value of $|X_1+X_2|$, given that the two roots $X_1, X_2$ of $X^2 + 2BX + 1 =0$ are real

$X^2 + 2BX + 1 =0$ The random variable B is normally distrubuted with mean zero and unit variance. Given that the two roots $X_1, X_2$ are real, find, giving your answers to three s.f. the ...
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### Gaussian distribution raised to a power

Given that $X$ follows a Gaussian distribution $e^{-x^2/2\sigma^2}$, what distribution is followed by $X^{1/3}$? How does one start to solve this problem? I guess it isn't ...
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### Solving for an unknown $\mu$ in a probability problem involving normal random variables.

(a): $P[X < 355] = P[Z < \frac{355 - 360}{4}] = P[Z < -1.25] = 1 - \Phi[1.25] = .1056$. Part (a) is simple, but I included it because I was not sure if I should somehow use it to solve ...
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### Determine $A$ such that $Q=X'AX$ has chi-squared distribution.

Let $\boldsymbol X\sim N_n(\boldsymbol\mu,\boldsymbol\Sigma)$, where $\boldsymbol\Sigma$ positive-definite. I am trying to determine, in general, what form $\boldsymbol A$ (one example is ...
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### Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
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### How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
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### $P(X=c)=0$ for normally distributed $X$?

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
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### statistics z score

I'm curious what the phrase "on average" means. Here is an example: On average, 30% were further than ___ kilometers away when they had their accident. Is $30\%$ a $z$-score or is it a mean? ...
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### if $X,Y$ i.i.d $\mathcal{N}(0,1)$, then $X+Y$ is independent of $X-Y$

I found on another thread* that if $(X+Y)$ is independent $(X-Y)$, and if $X,Y$ are i.i.d., then $X,Y$ are $\mathcal{N}(0,1)$ distributed. Is also the opposite true? Being $X,Y$ i.i.d ...