Discrete Mathematics for Computer Science

Exercises 529

(c) From part (b), with

X =(XI + X 2 + + X.)

and k = n, we have that

E(X1 + X2-+Xn) n' nE(X + X 2 + ' + X.)

so by part (a), this is

G)n

E E(Xi) 0

i=1

W Exercises

1. Consider a game based on the days of a 31-day month. A day is chosen at random-
   say, by spinning a spinner. The prize is a number of dollars equal to the sum of the
   digits in the date of the chosen day. For example, choosing the 31 st of the month pays
   $3 + $1 = $4, as does choosing the fourth day of the month.
   (a) Set up the underlying sample space ý2 and its probability density, the value of
   which at (o gives the reward associated with w.
   (b) Define a random variable X(o) on Q2 with a value at (o that gives the reward
   associated with wo.
   (c) Set up a sample space Q^2 x consisting of the elements in the range of X, and give
   the probability distribution px on Q^2 x arising from X.
   (d) Determine P(X = 6).
   (e) Determine P(2 < X < 4) = P(co : 2 < X(wo) < 4).

(f) Determine P(X > 10) = P(a) : X(wo) > 10).

2. Consider the following darts game: The target consists of a bull's-eye, which is a circle
   of radius 1, surrounded by a middle ring of outer radius 3 and inner radius 1; this region
   is in turn surrounded by another ring of outer radius 5 and inner radius 3. If you hit the
   bull's-eye, you win $10 plus the opportunity to throw again. If you hit the middle ring,
   you lose $2, and if you hit the outer ring, you lose $5. You must stop throwing as soon
   as you make a losing toss or hit three bull's-eyes. Suppose the probability that you hit
   a region is proportional to its area.
   (a) Set up an underlying sample space Q2 and its probability density p.
   (b) Define a random variable X on Q2.
   (c) Define a sample space Q^2 x and a probability distribution px on Q^2 x.


3. Suppose we flip a fair coin four times. We are interested in counting the number of
   times the coin turns up heads.
   (a) Define a sample space Q2 and a probability density p on Q2.
   (b) Define a random variable X on Q2 to count the number of heads.
   (c) Describe the event (X = 3) as a subset of QŽ.
   (d) Set up a sample space Q^2 x and a probability distribution px based on X. You may
   express your answer in terms of the binomial distribution.
   (e) Which number (or numbers) of heads is most likely to occur?