Approximations to Probability Distributions: Limit Theorems

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Approximations to Probability Distributions: Limit Theorems. Sequences of Random Variables. Interested in behavior of functions of random variables such as means, variances, proportions For large samples, exact distributions can be difficult/impossible to obtain

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### Approximations to Probability Distributions: Limit Theorems

Sequences of Random Variables
• Interested in behavior of functions of random variables such as means, variances, proportions
• For large samples, exact distributions can be difficult/impossible to obtain
• Limit Theorems can be used to obtain properties of estimators as the sample sizes tend to infinity
• Convergence in Probability – Limit of an estimator
• Convergence in Distribution – Limit of a CDF
• Central Limit Theorem – Large Sample Distribution of the Sample Mean of a Random Sample
Convergence in Probability
• The sequence of random variables, X1,…,Xn, is said to converge in probability to the constant c, if for every e>0,
• Weak Law of Large Numbers (WLLN): Let X1,…,Xn be iid random variables with E(Xi)=m and V(Xi)=s2 < . Then the sample mean converges in probability to m:
Other Case/Rules
• Binomial Sample Proportions
• Useful Generalizations:
Convergence in Distribution
• Let Yn be a random variable with CDF Fn(y).
• Let Y be a random variable with CDF F(y).
• If the limit as n of Fn(y) equals F(y) for every point y where F(y) is continuous, then we say that Ynconverges in distribution to Y
• F(y) is called the limiting distribution function of Yn
• If Mn(t)=E(etYn) converges to M(t)=E(etY), then Yn converges in distribution to Y
Example – Binomial  Poisson
• Xn~Binomial(n,p) Let l=np  p=l/n
• Mn(t) = (pet + (1-p))n = (1+p(et-1))n = (1+l(et-1)/n)n
• Aside: limn (1+a/n)n = ea
•  limn Mn(t) = limn (1+l(et-1)/n)n = exp(l(et-1))
• exp(l(et-1)) ≡ MGF of Poisson(l)
•  Xn converges in distribution to Poisson(l=np)
Central Limit Theorem
• Let X1,X2,…,Xn be a sequence of independently and identically distributed random variables with finite mean m, and finite variance s2. Then:
• Thus the limiting distribution of the sample mean is a normal distribution, regardless of the distribution of the individual measurements
Proof of Central Limit Theorem (I)
• Additional Assumptions for this Proof:
• The moment-generating function of X, MX(t), exists in a neighborhood of 0 (for all |t|<h, h>0).
• The third derivative of the MGF is bounded in a neighborhood of 0 (M(3)(t) ≤ B< for all |t|<h, h>0).
• Elements of Proof
• Work with Yi=(Xi-m)/s
• Use Taylor’s Theorem (Lagrange Form)
• Calculus Result: limn[1+(an/n)]n = ea if limnan=a