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

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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%.

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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%.

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A multivariate distribution:

A) specifies the probabilities associated with groups of random variables.

B) applies only to binomial distributions.

C) gives multiple probabilities for the same outcome.

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This is the definition of a multivariate distribution.

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Which of the following would least likely be categorized as a multivariate distribution?

A) The days a stock traded and the days it did not trade.

B) The return of a stock and the return of the DJIA.

C) The returns of the stocks in the DJIA.

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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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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.

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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.

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In a multivariate normal distribution, a correlation tells the:

A) strength of the linear relationship between two of the variables.

B) relationship between the means and variances of the variables.

C) overall relationship between all the variables.

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This is true by definition. The correlation only applies to two variables at a time.

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A multivariate distribution is best defined as describing the behavior of:

A) two or more independent random variables.

B) a random variable with more than two possible outcomes.

C) two or more dependent random variables.

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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.

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A normal distribution can be completely described by its:

A) skewness and kurtosis.

B) mean and mode.

C) mean and variance.

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

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A normal distribution is completely described by its:

A) mean, mode, and skewness.

B) variance and mean.

C) median and mode.

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

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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.

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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).

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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%.

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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).