Let `X` be a continuous random variable with probability density function
`f(x) = {(−4xlog_e(x),0<x<=1),(0,text(elsewhere)):}`
Part of the graph of `f` is shown below. The graph has a turning point at `x = 1/e`.
- Show by differentiation that `(x^k)/(k^2)(k log_e(x) - 1)` is an antiderivative of `x^(k – 1) log_e(x)`, where `k` is a positive real number. (2 marks)
- i. Calculate `text(Pr)(X > 1/e)`. (2 marks)
- ii. Hence, explain whether the median of `X` is greater than or less than `1/e`, given that `e > 5/2`. (2 marks)