Module 14: Central Limit Theorem

Question 1

formula for sample sum

Answer 1

Y = SUM from i = 1 to n (X_i)

Question 2

what is the formula for sample mean?

Answer 2

x̄ = 1/n SUM from i = 1 to n (X_i)

Question 3

what is the central limit thrm informally/conceptually?

Answer 3

As the sample size increases, the smapling distribution of both the sampl sum and sample mean get closer and closer to the normal distribution

Question 4

what is the central limit thrm mathematically?

Answer 4

Let X_1, X_2, … , X_n be indpeendent and identically distributed, random variables with finite mean µ and finite varaince σ^2. Then:

1. The sample sum converges in distribution to the normal distribution with mean nµ and varaince nσ^2
2. The sample mean converges in distributino to the normal distribution with mean µ and variance σ^2/n

Question 5

what is a contiuity correction?

Answer 5

a continuity correction is an adjustment that is made when a discrete random variable X is approixmated by a continuous random variable Y. It allows us to approximate the value P({X = i}) = P({i - 1/2 < Y < i +1/2})

Question 6

what is the DeMoivre-Laplace Approximation to the Binomial?

Answer 6

If X_n is a binomial random variable with parameters n, p and CDF F_n such that:

limit of F_n as n approaches infinity = F(x)

where F is the CDF of a normal random variable with mean np and variance np(1-p)