A farmer tests a new variety of wheat. They take a small sample of plants and count the seeds per ear: statistics Home Work MU, NZ

Problem 1

(a) (2 marks) Let X be random variables. Check that Var(X) E(X2) —

(b) (2 marks) Let X, y be random variables. Check that var(x + Y) Var(X) + Var(Y) + 2Cov(x, Y).

Problem 2

(a) (5 marks) A farmer tests a new variety of wheat. They take a small sample of plants and count the seeds per ear with the following result:
& = 41, & = 36, X3=25, & = 36, 36, & = 32, Sketch the empirical distribution function.
41, X6=32, & = 41, 25

(b) (1 mark) Do think the observed random variable is discrete or continuous? Give a reason for your answer.

(c) (5 marks) A hospital records for each birth the age of the mother over one year. The age is recorded in age groups instead of actual age. We assume that the distribution is uniform in each age group. Draw the corresponding distribution function over all age groups. Note that this will not be an empirical cdf.

(d) (1 mark) Is the random variable that corresponds to cdf in (iii) discrete or contin- nous? Give a reason for your answer.