## Solved – Limit of \$t\$-distribution as \$n\$ goes to infinity

I found in my intro to stats textbook that \$t\$-distribution approaches the standard normal as \$n\$ goes to infinity. The textbook gives the density for \$t\$-distribution as follows, \$\$f(t)=frac{Gammaleft(frac{n+1}{2}right)}{sqrt{npi}Gammaleft(frac{n}{2}right)}left(1+frac{t^2}{n}right)^{-frac{n+1}{2}}\$\$ I think it might be possible to show that this density converges (uniformly) to the density of normal as \$n\$ goes to infinity. Given \$\$lim_{nto infty}left(1+frac{t^2}{n}right)^{-frac{n+1}{2}}=e^{-frac{t^2}{2}}\$\$, … Read more

## Solved – Central Limit Theorem for square roots of sums of i.i.d. random variables

Intrigued by a question at math.stackexchange, and investigating it empirically, I am wondering about the following statement on the square-root of sums of i.i.d. random variables. Suppose \$X_1, X_2, ldots, X_n\$ are i.i.d. random variables with finite non-zero mean \$mu\$ and variance \$sigma^2\$, and \$displaystyle Y=sum_{i=1}^n X_i\$. The central limit theorem says \$displaystyle dfrac{Y – … Read more

## Solved – Can someone please explain to me what the particular scenarios mean

"The set of points in \$mathbb{R}^2\$ classified ORANGE corresponds to {\$x:x^Tβ>0.5\$}, indicated in Figure 2.1, and the two predicted classes are separated by the decision boundary {\$x:x^Tβ=0.5\$}, which is linear in this case. We see that for these data there are several misclassifications on both sides of the decision boundary. Perhaps our linear model is … Read more

## Solved – Can someone please explain to me what the particular scenarios mean

"The set of points in \$mathbb{R}^2\$ classified ORANGE corresponds to {\$x:x^Tβ>0.5\$}, indicated in Figure 2.1, and the two predicted classes are separated by the decision boundary {\$x:x^Tβ=0.5\$}, which is linear in this case. We see that for these data there are several misclassifications on both sides of the decision boundary. Perhaps our linear model is … Read more

## Solved – Proving that (X,Y) is not bivariate normal

Let \$X sim N(0,1)\$ and \$Y=X\$ if \$|X|>c\$ and \$Y=-X\$ if \$|X|<c\$, for any \$c>0\$. I've already proved that \$Y sim N(0,1)\$. How do we prove that \$(X,Y)\$ is not a bivariate normal? I've tried proving that \$Cov(X,Y)=0\$ and because they are not independent, then we reached our desired conclusion. However, I only get \$Cov(X,Y)= … Read more

## Solved – The sample size applied to a non-normal distribution

I have a single variable that represents my population values (sample of data):  94.51 59.81 63.84 94.51 94.51 94.51 94.51 94.51 94.51 94.51  59.81 94.51 94.51 94.51 47.90 29.16 50.36 23.51 44.41 33.14  47.90 29.16 47.90 29.16 47.90 29.16 47.90 29.16 47.90 29.16 …  23.44 24.52 12.37 29.12 24.52 12.37 29.12 … Read more

## Solved – Error in normal approximation to a uniform sum distribution

One naive method for approximating a normal distribution is to add together perhaps \$100\$ IID random variables uniformly distributed on \$[0,1]\$, then recenter and rescale, relying on the Central Limit Theorem. (Side note: There are more accurate methods such as the Box–Muller transform.) The sum of IID \$U(0,1)\$ random variables is known as the uniform … Read more

## Solved – How to compute the probability associated with absurdly large Z-scores

Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these Z-scores are bogus. Huge Z-scores correspond to extremely low associated probabilities. The values of the associated probabilities are given on e.g. the normal … Read more

## Solved – How to compute the probability associated with absurdly large Z-scores

Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these Z-scores are bogus. Huge Z-scores correspond to extremely low associated probabilities. The values of the associated probabilities are given on e.g. the normal … Read more

## Solved – Continuity correction error when using normal distribution to estimate Poisson distribution

I am doing a self-study exercise which attempts to exemplify a case where the Normal Distribution is used to approximate the Poisson Distribution, since the population mean is more than 10. I understand that when using Normal Distribution to Approximate Poisson / Binomial Distributions, there is a need for Continuity Correction Error to be managed. … Read more