The lifetime (in days) of a certain type of organisms is considered to be a random variable Y with density fy(y) = 1 200 (20-y) for 0 < y < 20, and fy (y) = 0 otherwise. (A) [3 marks] Find the life expectancy (i.e. expected lifetime) of these organisms by using the basic definition of a mean. (B) [4 marks] Generally, the survival function of a random variable X is defined as Sx(x) = P(X > x) for xR. Note that, in the current context, this can be interpreted as the probability of an organism surviving beyond time x. Show that the survival function of Y is Sy (y) = 1 for y0, 400 (20-y)2 for 0 < y < 20, (C) [4 marks] Justify that, in the current context, 0 for y 20. 00 E(Y) = Sy (y) dy, and make use of this result to obtain the life expectancy of these organisms.