lecture_25

Tue Jul 10 09:02:51 CDT 2012

Notes for homework due Wednesday

p.228 # 11 converging problem. Assume [g(x)] is not a huge function, say [|g( x)| < e^{M x}] for som fixed [M > 0]

Types of Random Variables

• Bernoulli [\subseteq] Binomial [(n,p)]
  • [p( i) = {n \choose i} p^i (1 - p)^{n - i}]
  • [E[ X] = n p]
  • [Var( X) = n p (1 - p)]
• Poisson
  • [p( i) =\frac{ e^{-\lambda} \lambda^i}{ i!}]
  • [E[ X] = Var( X) = \lambda]
• Geometric
  • [p( i) = p (1 - p)^{i - 1}]
  • [E[ X] = \frac{ 1}{ p}]
  • [Var( X) = \frac{ 1 - p}{ p^2}]
• Uniform [\subseteq] Beta
• Normal
• Exponential [\subseteq] Gamma
• Cauchy
  • [f(x) = \frac{ 1}{ \pi (1 + x^2)}]

Also note:

• Stirling's Formula
  • [k! \sim k^{k + \frac{ 1}{ 2}} e^{-k} \sqrt{ 2 \pi}]