Chapter 5 Notes

©2009 by S. Gramlich

(updated 9/30/2014)

 

Probability Distribution

! = Important Note

! These Notes are not meant to replace Reading.  Read Chapter first.

 

5-2

Probability Distribution (PD) Requirements:

            1) x discrete

            2) 0 <= P(x) <= 1

            3) ∑P(x) = 1

 

if the PD meets the above requirements then proceed with finding the following parameters:

mean:  μ = ∑[x * P(x)]

variance:  σ2 = ∑[(x-μ)2 * P(x)]

standard deviation: σ = √σ2

! I am citing Formula 5-2 from text but may have to use Formula 5-3 instead to get same answer in HW

 

Range Rule of Thumb (Confidence Interval)  for finding interval of usual scores:

max = μ + 2*σ

min = μ - 2*σ

so any # outside of the usual interval [min, max] is considered unusual.

 

Expected Value= average over long run

E = ∑[x * P(x)]

 

5-3

Inequality Complements

Prob(X at least #) = P(X >= #) = P(#) + P(# + 1) + P(# + 2) +…+ P(n)

=1 – P(X not at least #) = 1 – P(X < #) = 1 – [P(0) + P(1) +…+ P(#-1)]

Prob(X more than #) = P(X > #) = P(# + 1) + P(# + 2) +…+ P(n)

                        =1 – P(X not more than #) = 1 – P(X <= #) = 1 – [P(0) + P(1) +…+P(#)]

Prob(X at most #) = P(X <= #) =  P(0) + P(1) +…+P(#)

                        = 1– P(X not at most #) = 1 – P(X >= #) = 1 – [P(#) + P(# + 1) + P(# + 2) +…+ P(n)]

Prob(X fewer than #) = P(X < #) = P(0) + P(1) +…+ P(#-1)

                        =1 – P(X not fewer than #) = 1 - P(X >= #) = 1 – [P(#) + P(# + 1) + P(# + 2) +…+ P(n)]

 

Binomial (Bernoulli) Probability Distribution

Meets requirements for PD and:

  1. fixed n
  2. n independent
  3. 2 outcomes (binary): Success and Failure
  4. p constant

 

P(X) = nCx * px * qn – x where p = Prob(success) and q = Prob(failure) = 1 – p

Or

Use Table A-1 if 2 <= n <= 15 and given p on table

 

5-4 Binomial

mean:  μ = np

variance:  σ2 = npq = μq

standard deviation: σ = √σ2

μ - 2*σ <= usual <= μ + 2*σ

 

5-5 Poisson

Meets requirements for PD and:

  1. rv X = # of occurrences over some interval  (i.e. time or distance)
  2. X random
  3. X independent
  4. X uniformly distributed over interval

 

P(X) = (μx * e-μ) / x!

mean= variance:  μ= σ2

standard deviation: σ = √ μ

 

TECHNOLOGY

using StatCrunch: 

Stat – Calculators – Binomial

Stat – Calculators – Hypergeometric (“n” = A + B, “m” = “k” = A)

Stat – Calculators – Poisson

 

EXCEL 2007 commands:

=BINOMDIST(x, n, p, True/False)

=POISSON(x, μ, True/False)

False for =

True for <=

==HYPGEOMDIST(x, n, A, A+B)