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Reading 9: Common Probability Distributions-LOS k 习题精选

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

LOS k: Determine the probability that a normally distributed random variable lies inside a given interval.

 

 

A stock portfolio has had a historical average annual return of 12% and a standard deviation of 20%. The returns are normally distributed. The range –27.2 to 51.2% describes a:

A)
95% confidence interval.
B)
68% confidence interval.
C)
99% confidence interval.


 

The upper limit of the range, 51.2%, is (51.2 ? 12) = 39.2 / 20 = 1.96 standard deviations above the mean of 12. The lower limit of the range is (12 ? (-27.2)) = 39.2 / 20 = 1.96 standard deviations below the mean of 12. A 95% confidence level is defined by a range 1.96 standard deviations above and below the mean.

A stock portfolio's returns are normally distributed. It has had a mean annual return of 25% with a standard deviation of 40%. The probability of a return between -41% and 91% is closest to:

A)
90%.
B)
65%.
C)
95%.


A 90% confidence level includes the range between plus and minus 1.65 standard deviations from the mean. (91 ? 25) / 40 = 1.65 and (-41 ? 25) / 40 = -1.65.

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For a normal distribution, what approximate percentage of the observations fall within ±3 standard deviation of the mean?

A)
66%.
B)
99%.
C)
95%.


For normal distributions, approximately 99% of the observations fall within ±3 standard deviations of the mean.

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The mean return of a portfolio is 20% and its standard deviation is 4%. The returns are normally distributed. Which of the following statements about this distribution are least accurate? The probability of receiving a return:

A)
in excess of 16% is 0.16.
B)
of less than 12% is 0.025.
C)
between 12% and 28% is 0.95.


The probability of receiving a return greater than 16% is calculated by adding the probability of a return between 16% and 20% (given a mean of 20% and a standard deviation of 4%, this interval is the left tail of one standard deviation from the mean, which includes 34% of the observations.) to the area from 20% and higher (which starts at the mean and increases to infinity and includes 50% of the observations.) The probability of a return greater than 16% is 34 + 50 = 84%.

Note: 0.16 is the probability of receiving a return less than 16%.

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