1.  The constant is needed to make the density integrate to 1.

2.  beta

3.  (i)

4.  

betamem <- function(alpha,beta,t, u) { 

    
    # lhs = P(W > t+u) / P(W > t)
    lhs <- (1 - pbeta(t+u,alpha,beta)) / (1 - pbeta(t,alpha,beta))
    rhs <- 1 - pbeta(u,alpha,beta)
    list(lhs=lhs,rhs=rhs)  # left-hand and right-hand sides
}
 
print(betamem(1,1.8,0.5,0.2))  # 0.40, 0.67 

5.

dpower <- function(gamma,k)  # k any pos. integer
{   
   cc <- gamma - 1  # don't use c, since c() is an R ftn
   cc * k^(-gamma)
}

print(dpower(2.0,3))  # 0.11