Negative Binomial

1
Negative Binomial

• The binomial distn counts the number of successes
  in a fixed of trials
• The negative binomial distn counts the of trial
  needed to get a fixed of successes
• Specify p and R successes

2
Negative Binomial

• Ex
• 1. Need 3 hits on target to sink it. How many
  shots are needed?
• 2. After making 4 more free throws I can go home.
  How many shots are needed?
• 3. Need 5 mids with certain characteristics. How
  many mids do I need to question to find the 5?

3
Negative Binomial

• PMF(x) Prob that X trials are needed to get R
  successes
• Note it is assumed that the last trial is a
  success
• Otherwise, the RV is not well defined
• PMF(x) Prob(R-1 successes in X-1
  trials)Prob(last trial is a success)

4
Negative Binomial

• CDF(x) Prob( trials required is ltX)
• Prob(gtR successes in X trials)
• ngbprob(r,p,lo,hi)
• Theoretically, we need at least R trials to get R
  successes
• ngbprob(r,p,lo,hi) is defined for smaller values
  of lo

5
Negative Binomial

• Ex X tosses are required to get 5 heads
• Prob(Xlt10) ngbprob(5,1/2,0,10)
• prob(at least 5 heads in 10 tosses)
• ????

6
Negative Binomial

• Prob(Xgt10)
• If we use ngbprob, what is the value for HI?
• No theoretical upper bound for trials
• Several approaches
• 1. Set HI to be quite large, like 99
• 2. Run again with even larger HI
• If same answer, then were done

7
Negative Binomial

• Prob(Xgt10)
• Prob(lt5 heads in 9 tosses)
• So we can compute from BPROB
• ngbprob(r,p,lo,hi)
• LO and HI are trials required

8
Negative Binomial

• Exercises
• 1. A bin contains many apples. 20 are Xlarge. I
  need 3 more Xlarge to finish my box. What is the
  prob Ill finish my box within the next 10
  apples?
• 2. What is the prob it will take me more than 20
  apples to finish?
• 3. I am 90 sure I will finish within the next
  how many apples?
• 4. Find the median of apples required.

9
Negative Binomial

• Inference
• For upper bound on p, what is an unlikely range
  for trials
• If p is high, it is unlikely that trials will
  be large
• For lower bound on p, it is unlikely that
  trials will be small

10
Negative Binomial

• Ex Suppose that 8 trials are required to get 5
  heads
• Lower bound for p
• Solve ngbprob(5,p,0,8)0.10
• 0.3446
• Upper bound for p
• Solve ngbprob(5,p,8,99)0.10
• 0.8304

11
Negative Binomial

• Can be very confusing
• 1. ALWAYS do sanity check
• 8 trials for 5 heads sounds like p is about 5/8
• LOWER bound is LESS
• UPPER bound is MORE
• 2. Compute both bounds

12
Negative Binomial

• Exercises
• 1. Suppose that 10 trials are required to get 4
  successes. Find a 90 lower bound for p.
• 2. Find a 90 upper bound.
• 3. Find a 95 upper bound.

13
Negative Binomial

• Compare to binomial
• Suppose 8 trials produced 5 successes
• Lower bound on p gt X5 or more
• 0.3446
• Upper bound on p-gt 5 or fewer
• 0.8531
• NOT the same upper bound
• But lower bound is the same!

14
Negative Binomial

• Not quite the same data
• For NB, last trial is a success
• For Bin, last trial may or may not be a success

15
Negative Binomial

• Exercises
• Treat previous exercises as binomial. Answer same
  questions and compare.

16
Negative Binomial

• P-values
• Suppose we wish to test H0 p1/2 vs Ha plt1/2
• We observe that 12 trials are required to get 3
  successes

17
Negative Binomial

• What would be more extreme?
• Needing more trials
• PV Prob( trials gt 12)
• ngbprob(3, ½, 12, 99)
• 1- ngbprob(3, ½, 0, 11)
• 0.0327
• So not likely that p1/2

18
Negative Binomial

• Exercises
• 1. Test H0 p1/3 vs Ha plt1/3 when 15 trials are
  needed to get 3 successes
• 2. Repeat when p1/4
• 3. Repeat 1,2 if 12 trials were needed to get 3
  successes
• Relate all 4 answers. Which p-values should be
  smaller?