2. The standard deviation of a symmetrical distribution is 3 and the value of fourth moment about mean is 243. Hence the distribution is Leptokurtic-Justify.?

Question

Answer

Solution:

Given, standard deviation, σ = 3

Fourth moment about mean, μ = 243

To determine whether the distribution is leptokurtic or not, we need to calculate the kurtosis coefficient.

Kurtosis coefficient:

Kurtosis is a measure of the peakedness of the distribution. The kurtosis coefficient is given by:

β2 = μ4/σ^4 - 3

where β2 is the kurtosis coefficient.

Substituting the given values, we get:

β2 = (243/3^4) - 3

β2 = 9 - 3

β2 = 6

Interpretation:

Since the kurtosis coefficient is greater than 3, i.e., β2 > 3, the distribution is leptokurtic.

Leptokurtic distribution:

A leptokurtic distribution is one in which the peak of the distribution is higher and sharper than that of a normal distribution, implying that the distribution has more data points clustered around the mean. In other words, the tails of a leptokurtic distribution are fatter than those of a normal distribution.

Justification:

Hence, based on the calculated kurtosis coefficient, we can conclude that the given distribution is leptokurtic.

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