Probability

    

Assignment 12 

1 ) "Using an unfair coin as a fair die") Suppose you have an unfair coin with a probability p   of coming up heads and 

probability (1−p) of coming up tails on any given flip. 

Define a procedure for picking a number from 1 to 6 as follows:
- First, flip the coin 4 times.
- ifIf the total number of heads out of the 4 flips is anything other than 2, start over from the previous step. 

- Assuming the number of heads is 2, report a number from 1 to 6 according to the table below: 

 

If the sequence is,,,,Report the numbers 

HHTT.                                                                                                     1
HTHT.                                                                                                     2
HTTH                                                                                                     3

THHT                                                                                                     4

THTH                                                                                                     5

TTHH                                                                                                     6

 

Show that this procedure generates the numbers 1 to 6 with equal probability.
What if anything does this have to do with sufficient statistics? 

2) Suppose a certain unknown proportion F   of voters support a particular political candidate in the 

upcoming election, and you are interested in making inferences about F . You ask a pollster to perform a 

survey of n   people at random from the population, ask each person whether they support the candidate, 

  

and report the sequence of their responses. Instead, the pollster just records the total number of people 

out of n n  who said that they support the candidate. Do you need to redo the survey? Why or why not? 

3) Is it important to have a sufficient statistic for a parameter in order to do Bayesian inference? Why or why not? 

Rubric 

3Q Rubric