Which of the following is the correct ordering of the numbers q0.05,2q0.5,q0.02 , which are quantiles of a standard Gaussian variable?
q0.02<2q0.5<q0.05
2q0.5<q0.05<q0.02
q0.05<2q0.5<q0.02
q0.05<q0.02<2q0.5
2q0.5<q0.05<q0.02
The correct ordering of the numbers q0.05, 2q0.5, q0.02, which are quantiles of a standard Gaussian variable, is:
q0.02 < q0.05 < 2q0.5
To determine the correct ordering of the numbers q0.05, 2q0.5, and q0.02, which are quantiles of a standard Gaussian (normal) variable, we need to understand what these quantiles represent.
In a standard Gaussian distribution, also known as a standard normal distribution, the 0.5 quantile (also called the median) divides the distribution into two equal halves, with 50% of the data falling below it and 50% above it.
The 0.05 and 0.02 quantiles represent points in the distribution that separate lower percentages of the data. In this case, q0.05 represents the value below which approximately 5% of the data falls, and q0.02 represents the value below which approximately 2% of the data falls.
Given this understanding, let's evaluate the given orderings:
1. q0.02 < 2q0.5 < q0.05:
This ordering suggests that q0.02 is the smallest, followed by 2q0.5, and then q0.05. However, this is incorrect because q0.05 should be smaller than 2q0.5.
2. 2q0.5 < q0.05 < q0.02:
This ordering suggests that 2q0.5 is the smallest, followed by q0.05, and then q0.02. This is also incorrect because 2q0.5 should be larger than q0.05.
3. q0.05 < 2q0.5 < q0.02:
This ordering suggests that q0.05 is the smallest, followed by 2q0.5, and then q0.02. This is correct because q0.05 represents a lower percentile value than both 2q0.5 and q0.02.
4. q0.05 < q0.02 < 2q0.5:
This ordering suggests that q0.05 is the smallest, followed by q0.02, and then 2q0.5. This is incorrect because 2q0.5 should be larger than both q0.05 and q0.02.
Therefore, the correct ordering is q0.05 < 2q0.5 < q0.02.