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<-seq(from=3 , to=55 , length=53)

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

p<-c(rep(1/53,53))

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

p<-c(rep(1/53,53))

> Bp<-B*p

> mean<-sum(Bp)

> mean

[1] 29

#variance

> BBp<-B*B*p

sB<-sum(BBp)

> sB

[1] 1075

variance<-sB-mean^2

> variance

[1] 234

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

> condition1<-B>=5

> condition2<-B<=10.5

> z<-c(condition1&condition2)

> probability<-length(z[z==TRUE])/length(B)

> probability

[1] 0.1132075

(e) Create an R vector to represent a log-uniform prior distribution, where the probabilities are proportional to 1/. Hint: Construct a vector of un-normalised probabilities first, then normalise it the same way as you would normalise a posterior. [1 mark]

(f) According to the log-uniform prior distribution, what are the values of the same probability as in part (d)? [1 mark]

(g) Find the expected value and variance for where the probabilities are proportional to 1/. [1 mark]

Hello I need help with (e), (f) and (g). Thanks!