Answer to Question #234694 in Statistics and Probability for dhruv

Question #234694

The first four moments of a distribution about the value ‘4’ of the variable are -1.5,17,-30 and 108. Find the moments about mean and calculate the coefficient of skewness and kurtosis


1
Expert's answer
2021-09-21T00:26:23-0400

Solution:

"\u03b2_n =M(\u03be\u2212M_\u03be)\\ \u2014\\ the\\ n-th\\ central\\ moment.\n\\\\1)M(\u03be\u22124)=\u22121.5\n\\\\M\u03be=4\u22121.5=2.5\n\\\\M(\u03be\u2212M\u03be)=0\\ \u2014\\ the\\ first\\ central\\ moment."

"2) M(\\xi-4)^{2}=17\n\\\\M(\\xi-M \\xi)^{2}=M \\xi^{2}-(M \\xi)^{2}\n\\\\M \\xi^{2}-4^{2}=17\n\\\\M \\xi^{2}=33"

"\\begin{aligned}\n&M(\\xi-M \\xi)^{2}=33-(2.5)^{2}=26.75-\\text { the second central moment. } \\\\\n&\\text { 3) } M(\\xi-4)^{3}=-30 \\\\\n&M(\\xi-M \\xi)^{3}=M\\left(\\xi^{3}-3 \\xi^{2} M \\xi+3 \\xi(M \\xi)^{2}-(M \\xi)^{3}\\right)=M \\xi^{3}-3 M \\xi^{2} M \\xi+ \\\\\n&3(M \\xi)^{3}-(M \\xi)^{3}=M \\xi^{3}-3 M \\xi^{2} M \\xi+2(M \\xi)^{3} \\\\\n&M(\\xi-4)^{3}=M \\xi^{3}-3 \\cdot 33 \\cdot 4+2 \\cdot 4^{3}=-30 \\\\\n&M \\xi^{3}=238\n\\end{aligned}"

"M(\\xi-M \\xi)^{3}=238-3 \\cdot 33 \\cdot 2.5+2 \\cdot(2.5)^{3}=21.75-\\text{ the third central moment}"

"4) M(\\xi-4)^{4}=108"

"\\begin{aligned}\n&M(\\xi-M \\xi)^{4}=M\\left(\\xi^{4}-4 \\xi^{3} M \\xi+6 \\xi^{2}(M \\xi)^{2}-4 \\xi(M \\xi)^{3}+(M \\xi)^{4}\\right)= \\\\\n&=M \\xi^{4}-4 M \\xi^{3} M \\xi+6 M \\xi^{2}(M \\xi)^{2}-4(M \\xi)^{4}+(M \\xi)^{4}= \\\\\n&=M \\xi^{4}-4 M \\xi^{3} M \\xi+6 M \\xi^{2}(M \\xi)^{2}-3(M \\xi)^{4} \\\\\n&M(\\xi-4)^{4}=M \\xi^{4}-4 \\cdot 238 \\cdot 4+6 \\cdot 33 \\cdot 4^{2}-3 \\cdot 4^{4}=108\n\\end{aligned}"

"\\begin{aligned}\n&M \\xi^{4}=1516 \\\\\n&M(\\xi-2.5)^{4}=1516-4 \\cdot 238 \\cdot 2.5+6 \\cdot 33 \\cdot(2.5)^{2}-3 \\cdot(2.5)^{4}=\n\\end{aligned}"

"256.3125 - \\text{the fourth central moment}.\n\n\\\\\\Rightarrow\\sigma \\approx 5.17"

"A=\\frac{M(\\xi-M \\xi)^{3}}{\\sigma^{3}}=\\frac{21.75}{138.35}=0.157- \\text{asymmetry of the distribution}. \\\\E=\\frac{M(\\xi-M \\xi)^{4}}{\\sigma^{4}}=\\frac{256.3125}{715.5625}=0.358- \\text{kurtosis of the distribution}."


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