标题: Reading 9: Common Probability Distributions-LOS j 习题精选 [打印本页]
作者: 1215 时间: 2011-3-4 14:06 标题: [2011]Session 3-Reading 9: Common Probability Distributions-LOS j 习题精选
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:
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%.
作者: 1215 时间: 2011-3-4 14:07
The lower limit of a normal distribution is:
By definition, a true normal distribution has a positive probability density function from negative to positive infinity.
作者: 1215 时间: 2011-3-4 14:07
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?
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.
作者: 1215 时间: 2011-3-4 14:08
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:
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).
作者: 1215 时间: 2011-3-4 14:08
If X has a normal distribution with μ = 100 and σ = 5, then there is approximately a 90% probability that:
100 +/- 1.65 (5) = 91.75 to 108.25 or P ( P(91.75 < X < 108.25).
作者: 1215 时间: 2011-3-4 14:08
Which of the following statements about a normal distribution is least accurate?
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The distribution is completely described by its mean and variance. | |
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Kurtosis is equal to 3. | |
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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).
作者: 1215 时间: 2011-3-4 14:09
A normal distribution can be completely described by its:
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A) |
skewness and kurtosis. | |
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The normal distribution can be completely described by its mean and variance.
作者: 1215 时间: 2011-3-4 14:09
A normal distribution is completely described by its:
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A) |
mean, mode, and skewness. | |
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By definition, a normal distribution is completely described by its mean and variance.
作者: 1215 时间: 2011-3-4 14:09
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:
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A) |
correlation coefficient of the three random variables. | |
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B) |
conditional probabilities among the three random variables. | |
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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).
作者: 1215 时间: 2011-3-4 14:10
In addition to the usual parameters that describe a normal distribution, to completely describe 10 random variables, a multivariate normal distribution requires knowing the:
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.
作者: 1215 时间: 2011-3-4 14:10
In a multivariate normal distribution, a correlation tells the:
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strength of the linear relationship between two of the variables. | |
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B) |
relationship between the means and variances of the variables. | |
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overall relationship between all the variables. | |
This is true by definition. The correlation only applies to two variables at a time.
作者: 1215 时间: 2011-3-4 14:10
A multivariate distribution is best defined as describing the behavior of:
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two or more independent random variables. | |
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a random variable with more than two possible outcomes. | |
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two or more dependent random variables. | |
A multivariate distribution describes the relationships between two or more random variables, when the behavior of each random variable is dependent on the others in some way.
作者: 1215 时间: 2011-3-4 14:11
A multivariate distribution:
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specifies the probabilities associated with groups of random variables. | |
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applies only to binomial distributions. | |
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gives multiple probabilities for the same outcome. | |
This is the definition of a multivariate distribution.
作者: 1215 时间: 2011-3-4 14:11
Which of the following would least likely be categorized as a multivariate distribution?
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The days a stock traded and the days it did not trade. | |
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B) |
The return of a stock and the return of the DJIA. | |
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C) |
The returns of the stocks in the DJIA. | |
The number of days a stock traded and did not trade describes only one random variable. Both of the other cases involve two or more random variables.
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