**Self Test 3 for Probability
and Statistics**

This test was constructed by Dr.
Carol Dahl and Arturo Vasquez based on the Schaum's Outline *Theory and
Problems of Probability and Statistics* by Murray R. Spiegel. If you need
more review refer also to this outline.

** 1. True
False.** A financial economist is analyzing the
return of a certain asset in a mineral commodity market. He realized that the
expected return of the mineral asset in one year depends on a discrete random
variable Z whose probability function is given by f(z) = (1/2)

** 2. True
False.** A petroleum engineer has built a
mathematical model to analyze the behavior of a certain oil well. The oil
production (in thousand barrels) is considered as a continuous random variable
Y which has the following probability density function:

According to this model, E(Y) =
500 barrels, E(Y^{2}) = 1/2.

** 3. True
False.** The number of failures of a certain
oil pipeline during a period of five years is represented by a random variable
X which has the following density function:

A mechanical engineer has determined that the number of oil spills (Y) depends of X according to the following function: Y = 2X + 10. The probability of having between 12 oil spills and 16 oil spills in five years is 30%. The expected number of oil spillsis 10. (Contributed by Arturo Vasquez).

** 4. True
False. **A crazy investor wants to risk some money by investing in an asset whose
value depends on the copper price. Let X a random variable which represents the
possible values of the copper price in one year. According to some mineral
economists who are advising the crazy investor, the copper price can achieve
six possible values during the year (100 $ per ton): {1, 2, 3, 4, 5, 6}. The
probability of each event is the same. Hence, P(x) = 1/6. If the mineral
economists believe that the value of the asset (Y) depends on the copper price
by means of the following function: Y = 2X

** 5. True
False**. A mining company has two ore deposits which contain copper and zinc.
Let

What is the expected quantity of copper and zinc that is recoverable? If the area of the deposit of copper is expanded 3 times and the area of the deposit of zinc is expanded 2 times due to a success in the mining activities in the zone, the engineers consider that the production of copper and zinc can increase by the same proportions: 3X and 2Y. What is the expected total quantity of mineral (the sum of the random variables with the new coefficients) which is recoverable from the mine site? The expected values of copper and zinc are 2,667 tons and 3,444 tons respectively. The expected total quantity of mineral after the expansion is 15,667 tons (Contributed by Arturo Vasquez).

Variance, Standard Deviation and Some Theorems on Variance

__6. True
False__. Two different machines in a factory are able to produce
from 1 ton to 6 tons of refined steel plates individually. The steel is made
and packaged in lots of 1 ton each to be shipped to the factory's customers. It
has been shown through a number of studies that the output of each machine
varies day to day randomly and that the performance of one machine is
independent of the other. The probability of any given output is equal each
day. (There is a 1 in 6 chance that there will be 1 ton of output, or 2 tons of
output, or . . . , or 6 tons of output). The variance (a) and the standard
deviation (b) of the total number of tons produced per day by both machines
are:

a. 91/6

b. (91/6)^{1/2}

(Contributed by Dustin Menger)

**7. True
False**. The set {23.73, 24.46, 24.29,
19.64, 17.65, and 17.53} represents OPEC ref. basket oil prices of the second
semester of 2001. (a) The first moment about the origin is 21.22 and (b) the
second moment about the mean is equal to 9.18 (Contributed by Maria Sanchez).

**8. True
False**

**Standardized Random Variables**

**9. True False**.
A mineral economist is analyzing a sample of ten observations of the cooper
price in the

Average U.S. Produce Cooper Price (cents per pound)

Source:

He wants to standardize the copper price in order to compare its evolution with that of other mineral prices in the last years so as to verify if there were certain regularities or co-movements among prices. The standardized series is:

(Contributed by Arturo Vasquez)

**Moments and Moment Generating Function**

**10. True False.
**An oil company has developed a smart model which describes the probability
of extracting certain quantities of oil (X) in a well (barrels per day). The
density function of this model is:

X is expressed in thousands of barrels per day. It is necessary to calculate the mean and standard deviation of oil production because the managers of the company want to know if the expected production satisfies the requirement of 1500 barrels per day, which is the production level to break even. The chief statistician suggests using a moment generating function to determine the mean and standard deviation. Using this approach, the expected production level is 1600 barrels per day and its standard deviation is 500 barrels per day. It is profitable to exploit the oil well (Contributed by Arturo Vasquez).

**Variance for Joint
Distribution, Covariance, and Correlation**

**11. True
False**. A senior mining engineer is
analyzing the presence of copper (main mineral) and silver (by-product) in a
certain geological formation. The amounts of copper and silver are random
variables U and V respectively. The joint probability function of the U and V
(in thousand of tons) is given by f(u,v) = b(2u+v), where u and v can assume
all integers such that 0 ≤ u ≤ 2 and 0 ≤ v ≤ 3, and
f(u,v) = 0 otherwise. The engineer computes the following statistic parameters
in order to report this information to the management of the mine site where he
is working:

(a) E(U) = 29/21

(b) E(V) =13/7

(c) E(UV) = 17/7

(d)
E(U^{2})= 17/7

(e)
E(V^{2}) = 32/7

(f) Var (U) = 230/441

(g) Var (V) = 55/49

(h) Cov(U,V) = -20/147

(i) r= -0.2103 (correlation coefficient)

(Contributed by Dustin Menger and Arturo Vasquez)

**12. True False.
**There are two mineral assets (X and Y) which returns (in US$) are
negatively associated along time. An investor is thinking about buying both
assets in order to diversify his portafolio, but he firstly wants to know the
variability of the sum of the two assets before buying them. If the mineral
stock market reports that the variance of X is 50, the variance of Y is 20, and
the covariance of X and Y is -27, the standard deviation is US$ 10.
(Contributed by Arturo Vasquez).

**13. True
False.** In problem 10, a senior mining engineer
was working with a joint probability function of two discrete random variables
U and V (amounts of cooper and silver respectively). The probability function
is given by f(u,v) = b(2u+v), where u and v can assume all integers such that 0
≤ u ≤ 2 and 0 ≤ v ≤ 3, and f(u,v) = 0 otherwise. The
conditional expectation of V given U=2 is E(V|U=2)=20/11. (Contributed by Dustin Menger)

**14. True
False. **A mineral mining operation faces the
following probabilities of finding copper (Cu) and/or molybdenum (Mo) in the
stated quantities:

The expected value of copper given that 2,000 tons of molybdenum has been discovered is 583.33 tons. The conditional standard deviation of molybdenum given that 750 tons of copper has been discovered is 431 tons (Contributed by Sara Russell).

**Chebyshev's
Inequality and Law of Large Numbers**

**15. True False.
**In an African country, a petroleum engineer is working with a mineral
economist in determining how many barrels a small oil well can produce. Let X a
random variable which represents the oil production. The engineer determined
that the well can produce an average of 33 barrels with a variance of 16.
However, the management of the oil company considers that a well should produce
between 23 and 43 so as to obtain profits. The mineral economist has received
the task to determine the probability that the oil well produces between the
expected minimum and maximum values required by the management. After carrying
out the analysis, the mineral economist determine that at least the well will
produce in this range with a probability of 21/25 = 84% (Contributed by Arturo
Vasquez).

**16. True False**.
A mineral economist has in a database a sequence of different mental prices P

**17. True
False. **The density function of a
continuous random variable Z, which represents the population's income of a
mineral country, is:

Z is expressed in thousand US$ dollars. (a) The mode is 1.73, (b) the median is 1.62, and (c) the comparison between the mode, median and mean is Mean<Median<Mode (Contributed by Dustin Menger).

**18. True
False. **In problem 16, we had the density function of the population's income of
a mineral country. What is the income of the first 10% of the population, the
25% of the population, and the 75% of the population? The values are: US$ 500,
US$ 1200, and US$ 2121 (Contributed by Arturo Vasquez).

**Other Measures of Dispersion**

**19. True False.
**In problem 16, we had the density
function of the population's income of a mineral country. The

**Moments, Skewness and Kurtosis**

**20. True
False. **The returns of a mineral asset and their probabilities are given in the
following table:

Returns |
2 |
4 |
6 |
8 |

Probabilities |
0.4 |
0.3 |
0.2 |
0.1 |

In order to make an investment decision about this asset is necessary to
know some important information about the probability distribution. The
distribution is symmetric and the returns tend to share the same frequency of
occurrence (Contributed by Arturo Vasquez).