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FRM一级每日一题【风险管理基础32-定量分析】连续概率分布 II

来源：高顿网校 2015-03-04

　　习题：

　　Exercise:

　　Which of the following statements is the most accurate about the relationship between a normal distribution and a Student’s t distribution that have the same mean and standard deviation?

　　A They have the same skewness and the same kurtosis.

　　B The Student’s t distribution has larger skewness and larger kurtosis.

　　C The Kurtosis of a Student’s t distribution converges to that of the normal distribution as the

　　number of degrees of freedom increases.

　　D The normal distribution is a good approximation for the Student’s t distribution when the

　　number of degrees of freedom is small.

　　解析：

　　Answer: C

　　Explanation: The two distributions have the same skewness of zero but the Student’s t has higher kurtosis. As the number of degrees of freedom increases, the Student’s t distribution converges to the normal distribution, so c. is the correct answer.

　　知识点：

　　Student’s t Distribution (t distribution)

　　If Z is a standard normal variable and U is a chi-square variable with k degrees of freedom, which is independent of Z, then the random variable X:

　　follows a t distribution with k degrees of freedom.

　　Properties：

　　l The t distribution is symmetrical around its mean, which is equal to zero.

　　l The variance of the t distribution for k > 2 is k/(k-2).

　　l For low values of k, the t distribution looks very similar to a standard normal distribution, except that it displays excess kurtosis. As k increases, this excess kurtosis decreases. As k approaches infinity, the t distribution converges to a standard normal distribution.
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