Statistics 2

Statistics 2

1. The Binomial Distribution

   • X~B(n,p)
   • P(X«x)
   • P(X=r) = ⁿCᵣ x pʳ x qᵍ
     • where g is n-r
   • Conditions
     • 1) There must be a fixed number (n) of trials
     • 2) The trials must be independent
     • 3) The trials must only have two outcomes ( success and failure )
     • 4) The probability (p) of success must remain constant for each trial
   • E(x) = n x p
   • σ² = np(1-p)
   • σ = √np(1-p)

2. The Poisson Distribution

   • X~Po(λ)
   • Conditions
     • 1) Events occur singly in a continuous interval of space or time
     • 2) Events occur independently of one another
     • 3) Events occur at a constant rate
   • The rate of... indicates the Poisson distribution
   • E(X) = λ
   • Var(X) = λ

3. Approximations

   • Using the Poisson Distribution as an approximation to the Binomial Distribution
     • If X~B(n,p) where n>50 and p<0.1 then X~Po(λ) where λ = np
   • Using the Normal Distribution as an approximation to the Binomial Distribution
     • when n is large and p is not close to 0 or 1
       • If X~B(n,p) then X~N(np,npq)
       • This works best if n>10 and p=1/2
     • Continuity Correction
       • P(X=3) = P(2.5<X<3.5)
       • P(X<4) = P(X<3.5)
       • P(X«4) = P(X<4.5)
       • P(X»4) = P(X>3.5)
       • P(X>4) = P(X>4.5)
   • Using the Normal Distribution as an approximation to the Poisson Distribution
     • When λ is quite large
     • If X~Po(λ) then X~N(λ,λ)

4. Continuous Random Variables

   • The area under the graph of a continuous random variable is 1
   • To be a pdf f(x)»0 for all values of x in the given range
   • ₐ⌠ᵇ f(x) dx = 1
   • The Cumulative Distribution Function
     • F(x) = ₐ⌠ˣ f(x) dx where a is the lower limit
     • 0«F(x)«1
     • f(x) = d F(x)/dx
     • The median: F(m) = 0.5
     • F(LQ) = 0.25
     • F(UQ) = 0.75
   • The Mode: f'(x) = 0
     • The mode is the maximum value of the PDF
   • E(X) = ⌠xf(x) dx
     • E(aX + b) = aE(X) + b
   • Var(X) = ⌠x²f(x) dx - μ²
     • Var(aX + b) = a²Var(X)

5. The Continuous Uniform Distribution

   • X~U(α, β)
   • f(x) = 1/β-α
   • E(X) = 1/2 (α + β)
   • Var(X) = 1/12 (β - α)²

6. Hypothesis Testing

   • A hypothesis is a statement made about the value of a population parameter that we wish to test by collecting evidence in the form of a sample
   • The null hypothesis, Ho is the hypothesis that we assume to be correct unless proved otherwise
   • The alternative hypothesis tells us about the value of the population parameter if our assumption is shown to be wrong
   • The critical region is the range of values of the test statistic that would lead to rejection of Ho
   • The significance level is the probability of rejecting Ho when it is true
   • The actual significance level of a test is the probability of rejecting Ho
   • A hypothesis test is a mathematical procedure to examine a value of a population parameter proposed by the null hypothesis compared to the alternative hypothesis

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