Subject : Probability and statistics

4. Splitting a circle. I have a circle of circumference l. I choose it + 1 points pm pm, , pn on the circle independently and uniformly at random, splitting the circle into n + 1 arcs. The arc which begins at pg. is called I'ar'c 0," and number the remaining arcs clockwise from that one. For example: 'Hint: use calculus. (a) (2 points) Let L; be the random variable giving the length of are i. What is E[L;]? (b) (2 points) Suppose now that I choose a points at random in the interval [0.1]. splitting the interval into n + 1 segments. Let S, be the length of the ith segment. What is EISI]? (c) (2 points) Back to the circle. Let (1.: be the random variable giving the length of the are between 91 and P). where we start at p; and go clockwise until we reach ,0; (note this are may contain more than one of the n + l arcs from part (a)). What is the distribution of {13? (d) (2 points) [mark out an are a of length 9 on my circumference-l circle. Now] choose a random point on the circle; what is the probability that it lands in a? (e) (4 points) You're given a coin whose heads probability q is distributed uniformly on the interval [0.1]. You ip the coin n - 1 times. and let X be the resulting number of heads. Show that PrIX = k] = foreach k E l0.....n 1}