Statistics

Statistics

1. Poisson Distribution

   • Mean & variance approximately equal
   • Events occur : randomly, independently, uniformly over a given period of time
   • P(X=r) = e^(-lambda) X (lambda^r / r factorial)
     • Lambda is mean (or expected value)
   • Sum of 2 or more poissons
     • If X has mean lambda & Y has mean u
       • X+Y has mean (lambda + u) provided they are independent
   • Binomial approximation
     • If X~B(n,p) where n is large, p is small, then lambda=np
       • X~Poisson(lambda)
       • n>50
       • p<0.1

2. Normal Distribution

   • Normal Curve
     • Symmetrical
     • Bell-shaped
   • z = x-u / o-
   • Continuity Correction
     • If P(X=n)
       • P(n-0.5 < X < n+0.5
     • If P(X>n)
       • P(X > n+0.5)
     • If P(X<<n)
       • P(X < n+0.5)
     • If P(X<n)
       • P(X < n-0.5)
     • If P(X>>n)
       • P(X > n-0.5)
     • Discrete data, poisson approximation, binomial approximation
   • Binomial Approximation
     • X~B(n,p)
     • n is large, p is not too small or too large
     • Mean = np
     • Variance = npq
     • X~N(np,npq)
   • Poisson Aionpproximation
     • X~Poisson(lambda)
     • Lambda is large enough for distribution to be relatively symmetrical (>10)
     • X~N(lambda,lambda)

3. Sampling & Hypothesis Testing

   • Hypothesis testing
     • Define X
     • Ho: u=a
     • H1: u=/=a u>a u<a
   • Standardising
     • Parent population
       • X~N(u,o^2)
         • z = x-u / o
     • Sample distribution
       • X(bar) ~ N(u, o^2/n)
         • z = x-u / o/root(n)
   • Estimated standard deviation
     • When o not known for parent population, use sample o if sample size >50
   • Contingency tables
     • If test statistic > C.V reject Ho
     • Degrees of freedom
       • (m-1)(n-1)
     • chi^2 = sum of (O-E)^2 / E
     • Expected value = row total X column total / sample size
     • Look for big or small values of chi squared
     • Ho : thing 1 & thing 2 are independent
     • H1 : thing 1 & thing 2 are not independent

4. Bivarate Data

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