1    Instructions

• There are 3 questions in this assignment. Answer all questions.

• The marks awarded to each part of a question are indicated.

• You must provide explanations and reasoning for all of your answers in order to receive marks. An- swers without justification and explanation receive a mark of zero.

• This is an individual assignment – i.e., not group work. You must hand in your own work. This means writing up your answers in your own words and reasoning. Failure to do so is a breach of integrity,

and will be sent to the disciplinary board for further action.

2    Questions

Question 1. (10 Marks in total). A lake contains g gudgeon and r roach. You catch a sample of size n , on the understanding that roach are returned to the lake after being recorded, whereas gudgeon are retained in a keep-net.

1. Find the probability that your sample includes k gudgeon. (4 Marks in total)

2.  Show that as r and g increase in such a way that g/(g + r) - p , the probability distribution tends to the binomial.(6 Marks in total)

Question 2. (12 Marks in total, 3 marks per part). Find the distribution functions of (i) Z+  = max(0, Z}, (ii) X −  = min(0, Z}, (iii) IZI, and (iv) -Z in terms of the distribution function G of the random variable

Z

.

Question 3. (8 Marks in total, 4 marks per part). For what values of β e R is the moment E ╱ Z『← finite, if Z has