Q1. The lower limit of a normal distribution is:

A)     negative one.

B)     negative infinity.

C)     zero.

Q2. 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)   0.750.

C)   0.977.

Q3. 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)  4.56%.

C)  2.28%.

Q4. If X has a normal distribution with μ = 100 and σ = 5, then there is approximately a 90% probability that:

A)   P(91.8 < X < 108.3).

B)   P(93.4 < X < 106.7).

C)   P(90.2 < X < 109.8).

Q5. Which of the following statements about a normal distribution is least accurate?

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

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

C)   Kurtosis is equal to 3.

答案和详解如下：

Q1. The lower limit of a normal distribution is:

A)     negative one.

B)     negative infinity.

C)     zero.

Correct answer is B)

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

Q2. 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)   0.750.

C)   0.977.

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.

Q3. 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)  4.56%.

C)  2.28%.

Correct answer is C)

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).

Q4. If X has a normal distribution with μ = 100 and σ = 5, then there is approximately a 90% probability that:

A)   P(91.8 < X < 108.3).

B)   P(93.4 < X < 106.7).

C)   P(90.2 < X < 109.8).

Correct answer is A)

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

Q5. Which of the following statements about a normal distribution is least accurate?

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

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

C)   Kurtosis is equal to 3.

Correct answer is A)

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).

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