20. Correct. The answer is false. To answer this question, it is necessary to
calculate the skewness and kurtosis coefficients:
Skewness = 
. It consists of the third moment around the mean μ
divided by the standard deviation powered to 3.
Kurtosis = 
. It consists of the fourth moment around the mean μ
divided by the standard deviation powered to 4.
To compute these
coefficients, it is firstly necessary to calculate the moments around the
origin:
i) First moment around the origin: E[X] =
μ = 2(0.4) + 4(0.3) + 6(0.2) + 8(0.1) = 4
ii) Second moment
around the origin: E[X2] = 22(0.4) + 42(0.3) +
62(0.2) + 82(0.1) = 20
iii) Third moment
around the origin: E[X3] = 23(0.4) + 43(0.3) +
63(0.2) + 83(0.1) = 116.8
iv)
Fourth moment around the
origin: E[X4] = 24(0.4) + 44(0.3) + 64(0.2)
+ 84(0.1) = 752.
Now, we have to
calculate the central moments:
a) First central
moment: E[X- μ] = 0
b) Second central
moment: E[(X - μ)2] = E[X2]
- μ2 = 20 - 42 = 4. This is the variance. The standard
deviation is: σ = 2.
c) Third central moment:
E[(X - μ)3]
= E[X3] - 3*E[X]*E[X2] + 2*E[X]3
E[(X - μ)3] =
116.8 - 3(4)(20) + 2(4)3 = 4.8
d) Fourth central moment:
E[(X - μ)4]
= E[X4] - 4*E[X]*E[X3] + 6*E[X]2E[X2]
- 3*E[X]4
E[(X - μ)4] = 752 - 4(4)(116.8)
+ 6(42)20 - 3(44) = 35.2
The skewness coefficient is: 4.8 / (23) = 0.60 implying
the distribution is skewed to the right. It is not symmetric.
The kurtosis
coefficient is: 35.2 / (24) = 2.2 implying the distribution has less
kurtosis than the normal distribution which has kurtosis coefficient of 3. It
means that the values of the distribution tend not to share the same frequency
of occurrence. A coefficient of 2.2 means that the distribution is leptokurtic. A
frequency function with coefficient of kurtosis greater than zero is said to be
leptokurtic. It is more peaked about the mode than the normal distribution.