[2008] Topic 5: Some Important Probability Distributions 相关习题

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AIM 1: Describe the key properties of the normal distribution, the standard normal distribution.

1、Which of the following statements about the normal probability distribution is most accurate?

A) The normal curve is asymmetrical about its mean.

B) The standardized normal distribution has a mean of zero and a standard deviation of 10.

C) Five percent of the normal curve probability falls more than outside two standard deviations from the mean. 

D) Sixty-eight percent of the area under the normal curve falls between 0 and +1 standard deviations above the mean.

aianjie

 

The correct answer is C

The normal curve is symmetrical with a mean of zero and a standard deviation of 1 with 34% of the area under the normal curve falling between 0 and +1 standard deviation above the mean. Ninety-five percent of the normal curve is within two standard deviations of the mean, so five percent of the normal curve falls outside two standard deviations from the mean.

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2、A food retailer has determined that the mean household income of her customers is $47,500 with a standard deviation of $12,500. She is trying to justify carrying a line of luxury food items that would appeal to households with incomes greater than $60,000. Based on her information and assuming that household incomes are normally distributed, what percentage of households in her customer base has incomes of $60,000 or more?

A) 5.00%.

B) 2.50%.

C) 34.13%.

D) 15.87%.

aianjie

 

The correct answer is D

Z = ($60,000 – $47,500) / $12,500 = 1.0

From the table of areas under the normal curve, 84.13% of observations lie to the left of +1 standard deviation of the mean. So, 100% – 84.13% = 15.87% with incomes of $60,000 or more.

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3、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) 0.750.

B) 0.950.

C) 0.977.

D) 1.000.

aianjie

 

The correct answer is C

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.

aianjie

 

4、Standardizing a normally distributed random variable requires the:

A) mean and the standard deviation.

B) mean, variance and skewness.

C) natural logarithm of X.

D) variance and kurtosis.

aianjie

 

The correct answer is A

All that is necessary is to know the mean and the variance. Subtracting the mean from the random variable and dividing the difference by the standard deviation standardizes the variable.

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5、An investment has a mean return of 15% and a standard deviation of returns equal to 10%. Which of the following statements is least accurate? The probability of obtaining a return:

A) greater than 35% is 0.025.

B) less than 5% is 0.16.

C) greater than 25% is 0.32.

D) between 5% and 25% is 0.68.

aianjie

 

The correct answer is C

Sixty-eight percent of all observations fall within +/- one standard deviation of the mean of a normal distribution. Given a mean of 15 and a standard deviation of 10, the probability of having an actual observation fall within one standard deviation, between 5 and 25, is 68%. The probability of an observation greater than 25 is half of the remaining 32%, or 16%. This is the same probability as an observation less than 5. Because 95% of all observations will fall within 20 of the mean, the probability of an actual observation being greater than 35 is half of the remaining 5%, or 2.5%.