Reading 4: common probability distributions

Question 1

Which of the following is least likely an example of a discrete random variable?
The number of stocks a person owns.
The time spent by a portfolio manager with a client.
The number of days it rains in a month in Iowa City.

Answer 1

Time is usually a continuous random variable; the others are discrete. (LOS 4.a)

Question 2

For a continuous random variable X, the probability of any single value of X is:
one.
zero.
determined by the cdf.

Answer 2

For a continuous distribution p(x) = 0 for all X; only ranges of value of X have positive probabilities. (LOS 4.a)

Question 3

The cdf of 5, or F(5) is:
0.17.
0.71.
0.88.

chart: x;p(x), 0:0.04, 1:0.11, 2:0.18, 3:0.24, 4:0.14, 5:0.17, 6:0.09, 7:0.03

Answer 3

(0.04 + 0.11 + 0.18 + 0.24 + 0.14 + 0.17) = 0.88 (LOS 4.b)

Question 4

The probability that X is greater than 3 is:
0.24.
0.43.
0.67.

chart: x;p(x), 0:0.04, 1:0.11, 2:0.18, 3:0.24, 4:0.14, 5:0.17, 6:0.09, 7:0.03

Answer 4

(0.14 + 0.17 + 0.09 + 0.03) = 0.43 (LOS 4.b)

Question 5

What is P(2 ≤ X ≤ 5)?
0.17.
0.38.
0.73.

chart: x;p(x), 0:0.04, 1:0.11, 2:0.18, 3:0.24, 4:0.14, 5:0.17, 6:0.09, 7:0.03

Answer 5

(0.18 + 0.24 + 0.14 + 0.17) = 0.73 (LOS 4.b)

Question 6

The expected value of the random variable X is:
3.35.
3.70.
5.47.

chart: x;p(x), 0:0.04, 1:0.11, 2:0.18, 3:0.24, 4:0.14, 5:0.17, 6:0.09, 7:0.03

Answer 6

0 + 1(0.11) + 2(0.18) + 3(0.24) + 4(0.14) + 5(0.17) + 6(0.09) + 7(0.03) = 3.35 (LOS 4.b)

Question 7

A continuous uniform distribution has the parameters a = 4 and b = 10. The F(20) is:
0.25.
0.50.
1.00

Answer 7

F(x) is the cumulative probability, P(x < 20) here. Because all the observations in this distribution are between 4 and 10, the probability of an outcome less than 20 is 100%. (LOS 4.d)

Question 8

Which of the following is least likely a condition of a binomial experiment?
There are only two trials.
The trials are independent.
If p is the probability of success, and q is the probability of failure, then p + q = 1.

Answer 8

There may be any number of independent trials, each with only two possible outcomes. (LOS 4.e)

Question 9

Which of the following statements least accurately describes the binomial distribution?
It is a discrete distribution.
The probability of an outcome of zero is zero.
The combination formula is used in computing probabilities.

Answer 9

With only two possible outcomes, there must be some positive probability for each. If this were not the case, the variable in question would not be a random variable, and a probability distribution would be meaningless. It does not matter if one of the possible outcomes happens to be zero. (LOS 4.e)

Question 10

A recent study indicated that 60% of all businesses have a fax machine. From the binomial probability distribution table, the probability that exactly four businesses will have a fax machine in a random selection of six businesses is:
0.138.
0.276.
0.311.

Answer 10

Success = having a fax machine. 6! / 4!(6 – 4)!^4(0.4)^6 – 4 = 15(0.1296)(0.16) = 0.311. (LOS 4.e)

Question 11

Ten percent of all college graduates hired stay with the same company for more than five years. In a random sample of six recently hired college graduates, the probability that exactly two will stay with the same company for more than five years is closest to:
0.098.
0.114.
0.185.

Answer 11

Success = staying for five years. 6! / 2!(6 – 2)!^2(0.90)^6 – 2 = 15(0.01)(0.656) = 0.0984. (LOS 4.e)

Question 12

Assume that 40% of candidates who sit for the CFA® examination pass it the first time. Of a random sample of 15 candidates who are sitting for the exam for the first time, what is the expected number of candidates that will pass?
0.375.
4.000.
6.000.

Answer 12

Success = passing the exam. Then, E(success) = np = 15 × 0.4 = 6. (LOS 4.e)

Question 13

A key property of a normal distribution is that it:
has zero skewness.
is asymmetrical.
has zero kurtosis.

Answer 13

Normal distributions are symmetrical (i.e., have zero skewness) and their kurtosis is equal to 3. (LOS 4.f)

Question 14

Which of the following parameters is necessary to describe a multivariate normal distribution?
Beta.
Correlation.
Degrees of freedom.

Answer 14

To describe a multivariate normal distribution, we must consider the correlations among the variables, as well as the means and variances of the variables. (LOS 4.g)

Question 15

For the standard normal distribution, the z-value gives the distance between the mean and a point in terms of:
the variance.
the standard deviation.
the center of the curve.

Answer 15

This is true by the formula for z. (LOS 4.i)

Question 16

For a standard normal distribution, F(0) is:
0.0.
0.1.
0.5.

Answer 16

By the symmetry of the z-distribution and F(0) = 0.5. Half the distribution lies on each side of the mean. (LOS 4.j)

Question 17

Given a threshold level of return of 4%, use Roy’s safety-first criterion to choose the optimal portfolio.
Portfolio A.
Portfolio B.
Portfolio C.

Portfolio: E(Rp),Thetap
Portfolio A: 5%,8% Portfolio B: 11%,21%
Portfolio C: 18%,40%

Answer 17

SFR = (18 – 4) / 40 = 0.35 is the largest value. (LOS 4.k)

Question 18

Given a threshold level of return of 0%, use Roy’s safety-first criterion to choose the optimal portfolio.
Portfolio A.
Portfolio B.
Portfolio C.

Portfolio: E(Rp),Thetap
Portfolio A: 5%,8% Portfolio B: 11%,21%
Portfolio C: 18%,40%

Answer 18

SFR = (5 – 0) / 8 = 0.625 is the largest value. (LOS 4.k)

Question 19

For a lognormal distribution:
the mean equals the median.
the probability of a negative outcome is zero.
the probability of a positive outcome is 50%.

Answer 19

A lognormally distributed variable is never negative. (LOS 4.l)

Question 20

If a stock’s initial price is $20 and its year-end price is $23, then its continuously compounded annual (stated) rate of return is:
13.64%.
13.98%.
15.00%.

Answer 20

ln(23 / 20) = 0.1398 (LOS 4.m)

Question 21

A stock doubled in value last year. Its continuously compounded return over the period was closest to:
18.2%.
69.3%.
100.0%.

Answer 21

ln(2) = 0.6931 (LOS 4.m)

Question 22

Which of the following is least likely a property of Student’s t-distribution?
As the degrees of freedom get larger, the variance approaches zero.
It is defined by a single parameter, the degrees of freedom, which is equal to n – 1.
It has more probability in the tails and less at the peak than a standard normal distribution.

Answer 22

As the degrees of freedom get larger, the t-distribution approaches the normal distribution. As the degrees of freedom fall, the peak of the t-distribution flattens and its tails get fatter (more probability in the tails—that’s why, all else the same, the critical t increases as the df decreases). (LOS 4.n)

Question 23

Which of the following statements about the F-distribution and chi-square distribution is least accurate? Both distributions:
are typically asymmetrical.
are bounded from below by zero.
have means that are less than their standard deviations.

Answer 23

There is no consistent relationship between the mean and standard deviation of the chi-square distribution or F-distribution. (LOS 4.o)