Answer to Question #188684 in Statistics and Probability for Antara Adhikary

Question

1. The mean, variance,  β₁ and β₂ of a distribution are 10, 15, 1 and 3. Find the four moments about origin.

Answer 1

Mean = 𝑚1

′ =10, variance = 𝑚2 =15, coefficient of skewness = 𝛾1 = +1, coefficient of kurtosis is 𝛽2 = 4.

First moment about the origin is

Mean = 𝑚1

′ = 10.

Next,

𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑚2 = 15=

∑ 𝑥

2

𝑛

− (

∑ 𝑥

𝑛

)

2

.

Second moment about the origin is

𝑚2

′ =

∑ 𝑥

2

𝑛

= 𝑚2 + 𝜇1

2 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 + (

∑ 𝑥

𝑛

)

2

= 15 + 102 = 𝟏𝟏7

Coefficient of skewness is +1:

𝛾1 = +1 =

𝑚3

𝑚2

3

2

→ 𝑚3 = 𝑚2

3

2 = 16

3

2 = 64

Coefficient of kurtosis is 4:

𝛽2 = 4

𝑚4

𝑚2

2 = 4 → 𝑚4 = 4(16)

2 = 1024

Third moment about the origin is

𝑚3

′ = 𝑚3 + 3𝑚2𝑚1

′ + 𝑚1

′ 3 = 64 + 3(16)(10) + 103 = 𝟏𝟓𝟒𝟒

Fourth moment about the origin is

𝑚4

′ = 𝑚4 + 4𝑚3𝑚1

′ + 6𝑚2𝑚2

′ + 𝑚1

′ 4 = 1024 + 4(64)(10) + 6(16)102 + 104 = 𝟐𝟑𝟏𝟖𝟒