Restriction endonuclease cut at specific DNA sequences. For example, EcoRI cuts at the DNA sequence 5' GAATCC 3', which means you would expect 1 cut per 4096 random bases. (As an aside there are four DNA bases, and the EcoRI cut site is 6 bases long so 4^6 = 4096.)

1. What is the probability of finding 8 EcoRI cut sites in a DNA length of 50,000 random bases? Provide the correct pmf and show all work.
2. What assumptions are you making about the data in order to use this pmf?
3. What do you know about the mean and the variance for this process?
4. In an engineered plasmid, the EcoRI cut sites are evenly spaced throughout the length of the plasmid. Can you model this the same as you did above? Why or why not?
5. What is the probability of finding greater than or equal to 6, but less than or equal to 18 EcoRI cut sites in a DNA length of 50,000 random bases? Provide the answer in terms of f(x) and F(x), but you do not need to provide a numerical answer!
6. What is the probability of more than 5000 bases between EcoRI cuts? Provide the correct pdf and show all work.
7. Why are the expected values for Q3.1 and Q3.6 different? You do not need to calculate the expected values, but what do they represent?