MATH 232: Engineering Mathematics
Problem Set 11, Final version
Due Date: Mon., Mar. 15, 2010
Day 1 (Thursday)
• Ch. 2 Supplemental Exercises (pp. 48–49): Do Exercises 54, 55, 56 and 58. In doing
Problem 54, type help(mean) in R to see if the mean() command has a trim switch.
• Read Sections 3.1–3.3 in MRH4e.
Day 2 (Friday)
• Read Section 3.7 in MRH4e. Note that while the pmf fX (x) of a discrete random
variable is defined as fX (x) = P(X = x), the cumulative distribution function (or
cdf) FX (x) is defined as FX (x) = P(X ≤ x). For X =“sum of two fair dice”,
FX (4.5) = P(X ≤ 4.5) = P(X = 2) + P(X = 3) + P(X = 4)
2
3
1
1
+
+
=
.
= fX (2) + fX (3) + fX (4) =
36 36 36
6
An example of a graph of a cdf (for a different random variable X) appears in Figure
3-10, p. 90.
• Section 3.7: Do Exercises 80, 81, 83 and 84. Note that, to “verify a function is a pmf”,
you must check that all its values are nonnegative, and they total to 1.
You might use certain R commands to your advantage here. For instance, if my
random variable X were the sum of two dice, the following would be useful.
> pmf = c(1:6, seq(5,1,-1)) / 36
> sum(pmf)
[1] 1
> x = 2:12
> plot(x, pmf, type=’h’)
# quick way to produce my probabilities
# graphs the pmf/probability distribution
> sum(pmf[1:6])
[1] 0.5833333
# gives cdf value at x=7, or any x in interval [7, 8)
> sum(pmf[4:6])
[1] 0.4166667
# gives probability of rolling a 5, 6 or 7
MATH 232
> sum(x * pmf)
[1] 7
Problem Set 11
2
# calculates E(X)
> sum(xˆ2 * pmf) - sum(x*pmf)ˆ2
[1] 5.833333
> cdf = cumsum(pmf)
> plot(x, cdf, type=’s’)
# calculates Var(X)
# plots (in stair-step fashion) the cdf
• Problem ?19:
(a) Suppose that four coins (a penny, nickel, dime and quarter) are tossed and the
face-up side of each is observed as heads or tails. How many equally likely
outcomes are there? List them.
(b) For each x = 0, 1, 2, 3, 4, 5, compute the probability that exactly x many heads
occurs in the toss of five coins.
• Problem ?20: Suppose that you roll 5 standard dice. Determine the probability that
all the dice are the same.
• Problem ?21: Many games use spinners rather than dice to initiate action. A
classic board game published by Cadaco-Ellis is “All-American Baseball.” The
game contains discs for each of a number of baseball players. The disk for Nellie
Fox (the great Chicago White Sox second baseman) is pictured below.
The disc is placed over a peg with a spinner mounted in the center of the circle. The
spinner is spun and comes to rest pointing to the one of the numbered areas. Each
PS11—Final version
MATH 232
Problem Set 11
3
number corresponds to the possible result of Nellie Fox batting. (For example, 1 is
a homerun and 14 is a flyout.)
(a) Why is it unreasonable to believe that all the numbered outcomes are equally
likely?
(b) Explain how one could use the idea of equal likelihood to predict the probability
that the spinner will land on the sector numbered 14 and then make an estimate
of this probability.
(Spinners with regions of unequal size are used heavily in the K–8 textbook series
Everyday Mathematics to introduce probability to younger children.)
• Problem ?22: The traditional dartboard is pictured below.
A dart that sticks in the board is scored as follows. There are 20 numbered sectors
each of which has a small outer ring, a small inner ring, and two larger areas. A dart
landing in the larger areas scores the number of the sector, in the outer ring scores
double the number of the sector, and in the inner ring scores triple the number of a
sector. The two circles near the center score 25 points (the outer one) and 50 points
(the inner one). Unlike the last problem, it does not seem that an equal likelihood
model could be used to compute the probability of a “triple 20.” Explain why not.
PS11—Final version