Reading 9: Common Probability Distributions-LOS j 习题精选 normal, center, between, 2011

1215

Session 3: Quantitative Methods: Application
Reading 9: Common Probability Distributions

LOS j: Explain the key properties of the normal distribution, distinguish between a univariate and a multivariate distribution, and explain the role of correlation in the multivariate normal distribution.

 

 

A group of investors wants to be sure to always earn at least a 5% rate of return on their investments. They are looking at an investment that has a normally distributed probability distribution with an expected rate of return of 10% and a standard deviation of 5%. The probability of meeting or exceeding the investors' desired return in any given year is closest to:

A) 34%.

B) 98%.

C) 84%.

---

 

The mean is 10% and the standard deviation is 5%. You want to know the probability of a return 5% or better. 10% - 5% = 5% , so 5% is one standard deviation less than the mean. Thirty-four percent of the observations are between the mean and one standard deviation on the down side. Fifty percent of the observations are greater than the mean. So the probability of a return 5% or higher is 34% + 50% = 84%.

1215

The lower limit of a normal distribution is:

A) negative one.

B) zero.

C) negative infinity.

---

By definition, a true normal distribution has a positive probability density function from negative to positive infinity.

1215

A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account?

A) 0.950.

B) 0.977.

C) 0.750.

---

Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ ? 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 ? 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.

1215

A portfolio manager is looking at an investment that has an expected annual return of 10% with a standard deviation of annual returns of 5%. Assuming the returns are approximately normally distributed, the probability that the return will exceed 20% in any given year is closest to:

A) 0.0%.

B) 2.28%.

C) 4.56%.

---

Given that the standard deviation is 5%, a 20% return is two standard deviations above the expected return of 10%. Assuming a normal distribution, the probability of getting a result more than two standard deviations above the expected return is 1 ? Prob(Z ≤ 2) = 1 ? 0.9772 = 0.228 or 2.28% (from the Z table).

1215

If X has a normal distribution with μ = 100 and σ = 5, then there is approximately a 90% probability that:

A) P(93.4 < X < 106.7).

B) P(91.8 < X < 108.3).

C) P(90.2 < X < 109.8).

---

100 +/- 1.65 (5) = 91.75 to 108.25 or P ( P(91.75 < X < 108.25).

1215

Which of the following statements about a normal distribution is least accurate?

A) The distribution is completely described by its mean and variance.

B) Kurtosis is equal to 3.

C) Approximately 34% of the observations fall within plus or minus one standard deviation of the mean.

---

Approximately 68% of the observations fall within one standard deviation of the mean. Approximately 34% of the observations fall within the mean plus one standard deviation (or the mean minus one standard deviation).

1215

A normal distribution can be completely described by its:

A) skewness and kurtosis.

B) mean and mode.

C) mean and variance.

---

The normal distribution can be completely described by its mean and variance.

1215

A normal distribution is completely described by its:

A) mean, mode, and skewness.

B) variance and mean.

C) median and mode.

---

By definition, a normal distribution is completely described by its mean and variance.

1215

A multivariate normal distribution that includes three random variables can be completely described by the means and variances of each of the random variables and the:

A) correlation coefficient of the three random variables.

B) conditional probabilities among the three random variables.

C) correlations between each pair of random variables.

---

A multivariate normal distribution that includes three random variables can be completely described by the means and variances of each of the random variables and the correlations between each pair of random variables. Correlation measures the strength of the linear relationship between two random variables (thus, "the correlation coefficient of the three random variables" is inaccurate).

1215

In addition to the usual parameters that describe a normal distribution, to completely describe 10 random variables, a multivariate normal distribution requires knowing the:

A) 45 correlations.

B) overall correlation.

C) 10 correlations.

---

The number of correlations in a multivariate normal distribution of n variables is computed by the formula ((n) × (n-1)) / 2, in this case (10 × 9) / 2 = 45.