1. Ingeniería

4CCM141A/5CCM141B – Probability and Statistics I
Exercise Sheet 4
1. Four buses carrying 148 students from the same school arrive at a football stadium. The
buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly
selected. Let X denote the number of students that were on the bus carrying the randomly
selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the
number of students on her bus.
a.
Which of E(X) or E(Y) do you think is larger? Why?
b.
Compute E(X) and E(Y).
c.
Compute
2. If E(X)=1 and
and
.
, find: a.
b.
3. Consider the following gambling game, known as the wheel of fortune: A player bets on
one of the numbers 1 through 6. Three dices are then rolled, and if the number bet by the
player appears i times
then the player wins i units. If the number bet by the
player does not appear on any of the dice, then the player loses 1 unit. Is this game fair to
the player?
4. (Textbook 3.41, 3.42)
A multiple-choice examination has 15 questions, each with 5 possible answers, only one of
which is correct. Suppose that one of the students who take the examination answers each
of the questions with an independent random guess.
a. What is the probability that she answers at least ten questions correctly?
b. What is the expected value of correct answers? The standard deviation?
c. How does you answer to a. change if for each question, the student can correctly
eliminate one of the wrong answers?
d. How does you answer to a. change if for each question, the student can correctly
eliminate two of the wrong answers?
5. Let X be a Binomial random variable with parameters n and p. Show that
6. Suppose that X is a uniform discrete random variable between a and b
.
a. Write the probability mass function of X.
b. Calculate the expected value
and the standard deviation
of X.
7. In a certain TV show a fair die is tossed once. The participant gets a spherical container
full of perfume. The radius of the sphere is the result of the die measured in centimeters.
What is the average volume of perfume that participant gets?
(Hint: the volume of a sphere with radium R is
)
8. (Textbook 3.70, 3.82)
An oil prospector will drill a succession of holes in a given area to find a productive well.
The probability that he is successful on a given trial is 0.2.
a. What is the probability that the 3rd hole drilled is the first to yield a productive well?
b. If the prospector can afford to drill at most ten wells, what is the probability that
he will fail to find a productive well?
c. How many holes would the prospector expect to drill? Interpret your answer
intuitively.
9. (Partly based on Textbook 3.71.)
Let Y denote a geometric discrete random variable with probability of success p.
a. Calculate
.
b. Show that for a positive integer
:
c. Show that for positive integers
and
.
,
Why do you think this property is called the memoryless property of the geometric
distribution? Try to give a simple explanation for this result.
.
10. There are 12 fish in a pond, 3 large and 9 small. If you catch a small fish, you throw
it back. If you catch a large fish, you keep it. On average, how many fish must you catch to
get a large fish? (Assume that all fish are equally likely to be caught).
Optional exercises from textbook
3.15, (3.7+3.17), 3.24, 3.31, 3.36, 3.38, 3.44, 3.50, 3.61, 3.77, 3.83, 3.84