The second moment about mean oftwo distributions are 13.76 and 63.0 while the fourth moment about the men are 528.06 and 9500 respectively which of distribution is 1 leptokuric [2] mesokuric [3] platykurtic
To determine the kurtosis of a distribution based on the given information, we need to calculate the excess kurtosis coefficient. The excess kurtosis coefficient is the fourth moment about the mean divided by the square of the second moment about the mean, minus 3.
Let's calculate the excess kurtosis coefficient for each distribution:
For the first distribution:
- Second moment about the mean = 13.76
- Fourth moment about the mean = 528.06
Excess kurtosis coefficient = (Fourth moment about the mean / (Second moment about the mean)^2) - 3
= (528.06 / (13.76)^2) - 3
= (528.06 / 189.3376) - 3
= 2.788 - 3
= -0.212
For the second distribution:
- Second moment about the mean = 63.0
- Fourth moment about the mean = 9500
Excess kurtosis coefficient = (Fourth moment about the mean / (Second moment about the mean)^2) - 3
= (9500 / (63.0)^2) - 3
= (9500 / 3969.0) - 3
= 2.396 - 3
= -0.604
Now let's interpret the results:
- If the excess kurtosis coefficient is greater than 0, then the distribution is leptokurtic (heavy-tailed).
- If the excess kurtosis coefficient is equal to 0, then the distribution is mesokurtic (medium-tailed).
- If the excess kurtosis coefficient is less than 0, then the distribution is platykurtic (light-tailed).
Based on the calculations:
- The first distribution has an excess kurtosis coefficient of -0.212, so it is platykurtic.
- The second distribution has an excess kurtosis coefficient of -0.604, so it is also platykurtic.
Therefore, both of the distributions are platykurtic.