Question

1. Give the experimental line a real test. Come up with an n so that if the experimental line produces n chips with failure rate 6/38 or less, then the probability of getting a failure rate 6/38 or less under the original production system is less than 0.01.

2. If two random variables have the same generating function, must they have the same cumulative distribution function?

L.9) Central Limit Theorem Central Limit Theorem Version 1 says Go with independent random variables (X1, X2, X3, ..., Xn, ...] all with the same cumulative distribution function so that: Expect[X]ExpectX] and σ. Var[Xi]-Var[X] for all i and j Put: s[n] = As n gets large, the cumulative distribution function of S[n] is well approximated by the Normal[0, 1] cumulative distribution function. Another version of the Central Limit Theorem used often in statistics says Go with independent random variables sX1. X2, X3, ..., Xn, ...] all with the same cumulative distribution function so that: μ = Expect [Xi] = Expect [X] and σ' = Var[Xi]=Var[X] for all i and j Put SampleAverage[n]- As n gets large, the cumulative distribution function of SampleAverage[n] is wel Explain how the second version is a direct consequence of the first version. ated by the Normal , Cumulative distribution function. ▼ L.10) Gut version of the Central Limit Theorem from The Cartoon Guide to Statistics Comment on this sentiment found in the delightful book by Larry Gonick and Woollcott Smith: The Cartoon Guide to Statistics, HarperCollins, New York, 1993, p 83 ISBN: 0-06-273102-5 (pbk.) "Data that are influenced by small and unrelated random effects are approximately normally distributed. This explains why the normal is everywhere"

Answer 1

Answer

If a data, say X, is the cumulative effect of a number of small and unrelated random effects, say X1, X2, ……, Xn, then X = Σ(i = 1 to n)Xi. So, by CLT, X ~ N(μ, σ²), irrespective of the distribution of Xi’s.

In fact, it is this phenomenon, which is the underlying principle, behind the fact that majority of measurable variables are Normally distributed. Because, any measurable characteristic can be conceived of as a sum of the measurements of a number of smaller segments of the original characteristic and so by CLT, the measurable characteristic is Normally distributed.

For example, a pencil can be imagined to be a joint of a number of smaller pieces of pencils and so the pencil length is the aggregate of the length of all these smaller pencils.

DONE