Financial Risk Manager Part 1

Explanation

The correct answer is C because there is no consistent relationship between the mean and standard deviation in either the F-distribution or the chi-square distribution.

Analysis of each option:

A. Both are asymmetrical - This is accurate. Both F-distribution and chi-square distribution are positively skewed (asymmetrical) distributions.

B. Both have a bound equal to zero on the left - This is accurate. Both distributions have a lower bound of zero (they cannot take negative values).

C. Their means are always less than their standard deviations - This is NOT accurate. There is no consistent relationship between the mean and standard deviation for these distributions. The relationship depends on the degrees of freedom parameters:

• For chi-square distribution with k degrees of freedom: Mean = k, Variance = 2k, Standard deviation = √(2k)
• For F-distribution with d₁ and d₂ degrees of freedom: Mean = d₂/(d₂-2) for d₂ > 2, Variance is more complex

The mean may be greater than, less than, or equal to the standard deviation depending on parameter values.

D. They are defined by the number of degrees of freedom - This is accurate. Both distributions are parameterized by degrees of freedom:

• Chi-square: single parameter (degrees of freedom)
• F-distribution: two parameters (numerator and denominator degrees of freedom)

Key Points:

1. Both distributions are asymmetric and bounded at zero.
2. Both are defined by degrees of freedom parameters.
3. The relationship between mean and standard deviation is not fixed and depends on parameter values.
4. The statement in option C makes an absolute claim ("always") that is not supported by the mathematical properties of these distributions.