a)Letbe a fixed positive integer. Let be a random variable that has possible values{1,2,...,}. Consider the probabilities

F(m)=P(Xm)for m1. It's a good idea to draw a number line and color the event {Xm}for a genericm. For1kN, writeP(X=k)in terms of the valuesF(m)for1

m1.

b)LetX1,X2,...,Xnbe the results ofndraws made at random with replacement from {1,2,...,N}. LetM=max{X1,X2,...,Xn}. Use the method developed in Partato find the distribution ofM.

[Think about howMcan be at mostm. For this to happen, how big canX1be? What aboutX2? If you have trouble starting out in the general case, pick some small numbers like N=10,m=4, andn=3to see what's going on.]

c)Now letX1,X2,...,Xnbe the results ofndraws made at random without replacement from{1,2,...,N}. You can assumenNin this case. LetM=max{X1,X2,...,Xn}. Use the method developed in Partato find the distribution ofM. Start by carefully specifying the possible values ofM.