Chapter 4 Quiz Questions

Probability distribution of a discrete random variable X x 0 1 2 3 4 5 6 7 P(X) 0.04 .11 .18 .24 .14 .17 .09 .03 What is P(2 ≤ X ≤ 5)?

.073. (0.18 + 0.24 + 0.14 + 0.17) = 0.73

The expected value of the random variable X is:

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

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. Success = staying for five years. [6! / 2!(6 − 2)!](0.10)2(0.90)6-2 = 15(0.01)(0.656) = 0.0984.

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:

0.311. Success = having a fax machine. [6! / 4!(6 − 4)!](0.6)4(0.4)6-4 = 15(0.1296)(0.16) = 0.311.

Probability distribution of a discrete random variable X x 0 1 2 3 4 5 6 7 P(X) 0.04 .11 .18 .24 .14 .17 .09 .03 the probability that x is greater than 3 is

0.67. (0.14 + 0.17 + 0.09 + 0.03) = 0.43

Probability distribution of a discrete random variable X x 0 1 2 3 4 5 6 7 P(X) 0.04 .11 .18 .24 .14 .17 .09 .03 the cdf of 5, or F(5) is:

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

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

1.00. 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%

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:

13.98%. ln(23 / 20) = 0.1398 (LOS 4.m)

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?

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

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

69.3%. ln(2) = 0.6931 (LOS 4.m)

A study of hedge fund investors found that their annual household incomes are normally distributed with a mean of $175,000 and a standard deviation of $25,000. The percentage of hedge fund investors that have incomes greater than $150,000 is closest to:

84.13%. 1 − F(-1) = F(1) = 0.8413. There is an 84.13% probability that a randomly chosen income is not more than one standard deviation below the mean. (LOS 4.h)

Portfolio Portfolio A Portfolio B Portfolio C E(Rp) 5% 11% 18% σp 8% 21% 40% Given a threshold level of return of 0%, use Roy's safety-first criterion to choose the optimal portfolio. Portfolio:

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

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

As the degrees of freedom get larger, the variance approaches zero. 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)

Which of the following is least likely a probability distribution? X = [1,2,3,4]; Prob [Xi] = Xi^2/30. X = [5,10]; Prob [Xi] = 8−Xi/5. X = [5,10]; Prob [Xi] = Xi−3/9.

B. 8−5/5+8−10/5=1/5, and 8−10/5 is negative, so this satisfies neither of the requirements for a probability distribution. The others have P[Xi] between zero and one and ∑P[Xi]∑P[Xi] = 1, and thus satisfy both requirements for a probability distribution.

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

B. time is usually a continuous random variable, the others are discrete

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

B: the probability of a negative outcome is zero . A lognormally distributed variable is never negative. (LOS 4.l)

For a standard normal distribution, F(0) is:

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)

Portfolio Portfolio A Portfolio B Portfolio C E(Rp) 5% 11% 18% σp 8% 21% 40% Given a threshold level of return of 4%, use Roy's safety-first criterion to choose the optimal portfolio. Portfolio:

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

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

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

Which of the following parameters is necessary to describe a multivariate normal distribution?

Correlation. 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)

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.

The probability of an outcome of zero is zero. 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

A key property of a normal distribution is that it:

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

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

the standard deviation.

Which of the following is least likely a condition of a binomial experiment? there are only 2 trials the trials are independent if p is the prob of success and q is the prob of failure, then p+q=1

there are only 2 trials. There may be any number of independent trials, each with only two possible outcomes.

For a continuous random variable X, the probability of any single value of X is:

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