AMS 5 Lecture 16: Class 16 - The Central Limit Theorem

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Applied Math and Statistics
Yonatan Katznelson

AMS 5 Lecture 16 5122017 (8:009:05) The Central Limit Theorem (CLT) n tickets are drawn at random with replacement from a box of tickets with avg = A and SD = s o Expected value of the sum EV(Sum) = n*A o Standard Error of the sum of the draws: SE(Sum) = n 12* s o Box of sums: tickets labeled with sums of all possible draws of n tickets from the average box. o Avg(Box of sums) = n * A = E U (Sum) o SD(Box of sums) = n 12* s = SE(Sum) CLT If n is large enough, then the box of sums has an approximately normal distribution Browninan Motion Botanist in 1950s who was studying jar of pollen, and in jar he saw the pollen was moving, and the movement looked random, however it didnt stray too far from the original observed area. Its moving because the water molecules are moving and striking the pollen in all directions, trying to force it to equilibrium. Example: Box has an average of 10 and a SD of 2. n = 10000 tickets are drawn at random with replacement from this box. o EV(Sum) = 10*10000 = 100,000 o SE(Sum) = (10000) *2 = 200 o Expect sum to be about 100000 + 200 (law of averages) o P(99600 <= obs sum <= 100400) = ? P( obs sum EV < 400) = P( obs Avg < 230) ~ 95 o More generally: P(A < obs sum < B) = Convert to standard units
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