A continuous random variable X has probability density function f(x). The probability that X x is given by the function F(x). Explain why F(x) = f(x). A rod of length 2a is broken into two parts at a point whose position is random. State the form of the probability distribution of the length of the smaller part, and state also the mean value of this length. Two equal rods, each of length 2a, are broken into two parts at points whose positions are random. X is the length of the shortest of the four parts thus obtained. Find the probability, F(x), that X x, where 0 < x a. Hence, or otherwise, show that the probability density function of X is given by f(x) = (2(a-x) a for 0 < x a for x 0, x> a 1 Show that the mean value of X is a. Write down the mean value of the sum of the two smaller parts and show that the mean values of the four parts are in the proportions 1:2:4:5. [JMB]