A02.1 Butterfly moments The lifetime, X, in days of a species of butterfly has a density of fx(x) =1 6/25 1 > 1 (You discovered the value of b= 4 ) (a) Find the expectation of X". What constraints are there on a for the expectation to be finite? (b) Write down the expectation and variance of X. [3] A02.2 More moments A random variable, X, has a density of 6r(1-r) for 0 -2. (c) Use the linearity of expectation to find E [X + X? + X+ X' + ...] (it is valid to switch the order of expectation and sum even though the series is infinite - if inter- ested, look up Fubini's Theorem). [7] A02.3 Challenge Question For a sufficiently smooth function, f(x), Taylor expan- sion about some point, /, gives f ( I ) = f ( H) + ( x - 1) f' ( 1 ) + 5 ( 2 - 12 ) 2 5" ( 1 ), for some n(, x) between / and r. Consider any function f with f"(x) 2 0 for all re R and show that E [f(X)] 2 f(E [X]). Hence relate E [X"?] to E [X] [3]