2. Sample Maximum

a)Let

Nbe a fixed positive integer. Let

Xbe a random variable that has possible values{1,2,...,}

{1,2,...,N}. Consider the probabilities()=()

F(m)=P(Xm)for1

m1. It's a good idea to draw a number line and color the event{}

{Xm}for a generic

m.

For1

1kN, write(=)

P(X=k)in terms of the values()

F(m)for1

m1. If you get stuck, take a look atExample 2.2.2in the textbook.

b)Let

1

,

2

,...,

X1,X2,...,Xnbe the results of

ndraws made at random with replacement from{1,2,...,}

{1,2,...,N}. Let=max{

1

,

2

,...,

}

M=max{X1,X2,...,Xn}. Use the method developed in Partato find the distribution of

M.

[Think about how

Mcan be at most

m. For this to happen, how big can

1

X1be? What about

2

X2? If you have trouble starting out in the general case, pick some small numbers like=10

N=10,=4

m=4, and=3

n=3to see what's going on.]

c)Now let

1

,

2

,...,

X1,X2,...,Xnbe the results of

ndraws made at random without replacement from{1,2,...,}

{1,2,...,N}. You can assume

nNin this case. Let=max{

1

,

2

,...,

}

M=max{X1,X2,...,Xn}. Use the method developed in Partato find the distribution of

M. Start by carefully specifying the possible values of

M.