approximations to probability distributions limit theorems l.
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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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sequences of random variables
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
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
Other Case/Rules
  • Binomial Sample Proportions
  • Useful Generalizations:
convergence in distribution
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
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
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
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