Please use R to answer this question and include the code you used with your answers. This question is about the 'log-uniform' prior distribution which we'll use at times during the course.

A quantity is known to be positive and somewhere between 3 and 55 (inclusive).

(a) In R, make a vector called beta that contains the set of possible values. Go from 3 to 55 in steps of 1. [1 mark]

(b) Create an R vector to represent a discrete uniform prior distribution, i.e., assigning equal prior probabilities to each possibility. [1 mark]

(c) Find the expected value and variance for . [2 marks]

(d) According to the uniform prior distribution, calculate P (5 10.5)?

Hint: In R, the element-wise logical and operation for vectors is &. [2 mark]