Download 1 Law of Large Numbers 2 Central Limit Theorem

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Law of Large Numbers
Definition: As a procedure is repeated again and again, the relative frequency probability
of an event:
P (A) =
number of times A occured
number of times trial was repeated
tends to approach the actual probability.
Example: In lab three we simulated coin flips by sampling from the Binomial(n = 1,p = .5).
I 1 represented a head and a 0 represented a tail. We know that the actually probability of
getting a head (call this event A) is 0.5, but we want to demonstrate the law of large numbers.
We draw a value from a Binomial(1,0.5) ten times and note how many ones came up. We
expect 5/10 ones, but not everyone got that value; some got only 1/10!!! However, when we
draw 10000 times from the Binomial(1,0.5), everyone was very near the actual probability of
0.5. Thus by repeating the procedure of drawing from the Binomial our probability frequency
of ones is getting closer to the actual probability!!!!
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Central Limit Theorem
(See lab for full definition or book section 5-5) Short Def.: Suppose the R.V. X has
theoretical mean E(X) = µ and theoretical standard deviation σ. We take repeated samples
of size n (where n is large) from this population, then the sample mean
x̄ ≈ N E(x̄) = µx̄ = µ, σx̄ = √σn
2.1
Standard Normal Distribution
If the population follows a standard normal distribution, then the theoretical mean is E(X) =
µ = 0 and theoretical standard deviation is σ = 1.
2.2
Uniform Distribution
If the population follows a Uniform distribution with minimum= a and maximum= b, then
b−a
and theoretical standard deviation is σ = √
(note
the theoretical mean is E(X) = µ = b+a
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this value is given to you in the lab.
2.3
Binomial Distribution
If the population follows a Binomial distribution with parameters n andp
p, then the theoretical mean is E(X) = µ = np and theoretical standard deviation is σ = n(1 − p)p
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