normal distribution

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

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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

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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

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"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

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"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 – The sample size applied to a non-normal distribution

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I have a single variable that represents my population values (sample of data): [1] 94.51 59.81 63.84 94.51 94.51 94.51 94.51 94.51 94.51 94.51 [11] 59.81 94.51 94.51 94.51 47.90 29.16 50.36 23.51 44.41 33.14 [21] 47.90 29.16 47.90 29.16 47.90 29.16 47.90 29.16 47.90 29.16 … [331] 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

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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

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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

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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

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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