Summation Examples

There are times when an infinite series must be evaluated. Some of these problems are easy to solve and others are more challenging.

1. [Exercise 2.3.4(e)] Let X equal the number of rolls of a pair of dice that are needed to determine whether a bettor wins or loses playing craps. The p.d.f. of X is f (1) = 12/36 and

f ( x ) = 2[(3/36)(9/36)(27/36)^( x -2) + (4/36)(10/36)(26/36)^( x -2) + (5/36)(11/36)(25/36)^( x -2)]

or

[Maple Math]

for x = 2, 3, ... . Show that

E ( X ) = [Maple Math]

and

Var( X ) = [Maple Math] .

Solution

2. [Exercise 3.6.11] Flip n fair coins until heads has been observed on each coin. E.g. toss the coins and remove the heads, toss the remaining coins and remove the heads, etc. Let [Maple Math] equal the number of tosses required. It can be shown that the p.d.f. of [Maple Math] is

[Maple Math] .

Find [Maple Math] . How does the value of n affect [Maple Math] ? Solution

3. Flip a coin successively. Let [Maple Math] equal the number of flips needed to observe

(a) [Exercises 1.1.6 and 3.6.20] The same face on successive flips.

(b) [Exercises 1.1.7 and 3.6.21] HH on successive flips.

(c) [Exercises 1.1.8 and 3.6.22] HT on successive flips. Solution

4. Let [Maple Math] have a Poisson distribution with mean [Maple Math] .

(a) [Exercise 3.7.5] Find the mean, variance, and other moments of [Maple Math] symbolically.

(b) [Exercise 6.1.1] Both [Maple Math] -bar and the sample variance, [Maple Math] , are unbiased estimators for [Maple Math] . Compare the variances of these two estimators both theoretically and empirically. Solution