# Tagged Questions

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

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### Generate a set of random numbers with an average evenly distributed between two given values

1) I generate 1000 random numbers between 0 and 10 and take the average. If I do the above action "many" times the resulting average values will be a normal distribution over 0 to 10. Correct? What ...
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### Why a normal distribution would not give a good approximation to the distribution of marks

An examination is marked out of $100$. It is taken by a large number of candidates. The mean mark, for all candidates, is $72.1$ and the standard deviation is $15.2$. Give a reason why a normal ...
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### How to calculate inverse cumulative distribution using a table?

I need help with this: $$P(X\geq a)=1-F_X(a)=1-\Phi\left(\frac{a-70}{8}\right)=0.25$$ When $X\sim N(70,64)$. I know that it should be: $(a-70)/8 = 0.6745$ How do I get $0.6745$ From $Z$ table? I ...
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### $\lim X_n = 0$ iff $b > 0$

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
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### $\frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx$ - Normal Distribuition

I have read in one of my finance books (Asset Pricing - John H. Cochrane) that there is this identity: \begin{split} \frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx &...
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### Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
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### Check if $Cov(X_1+X_2,X_1-X_2)=0$, i.e. if independent?

"Let $X_1$ and $X_2$ be independent, $N(0,1)$-distributed random variables. Show that $X_1+X_2$ and $X_1-X_2$ are independent." I know that for multivariate normal distributions independence can be ...
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### Variance of a normal distribution for coin toss.

I have difficulties constructing the normal distribution for (20) coin tosses. (Don't ask why, but I never had probability in school.) What is the probability of getting at most 12 heads out of 20 ...
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### how to find the cdf of X in terms of Z when $X=2Z+1$

Consider Z a Normal (Gaussian) random variable with mean 0 and variance 1. It has density $$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} \text{for all x real numbers}$$ We consider $X=2Z+1$. Write ...
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### Probability more than 25% greater?

The random variable X is distributed N(60,64). The random variable Y is distributed N(52,36). Find the probability that a random observation from X is more than 25% greater than a random observation ...
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### Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
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### Normal distribution without standard deviation given

The proportion of pink candies in a bag is supposed to be $50\%$. The filling machine is to be tested to see if it fills with the right proportion. A random sample of $50$ candies is made. The machine ...
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### 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 $N(a\mu,\sigma)$...
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### Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho$ for all $i\neq j$. I'd like to compute E\left(\sum_{i=1}^nX_i\right)^4. ...
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### Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
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### How to use Chebyshev Inequality

Use Chebyshev Inequality to estimate the probability that in any one day of a business that earns a mean of 100 dollars a day with a standard deviation of 28.87 dollars, that business will make either ...
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### 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 ...