Relation between moments about Mean and Arbitrary point
The 𝑟th moment about mean is given by
𝑚_𝑟=1/𝑛 ∑_(𝑖=1)^𝑛▒〖𝑓_𝑖 (𝑥_𝑖−𝑥 ̅ )〗^𝑟 ; where r = 0, 1, ...
=1/𝑛 ∑_(𝑖=1)^𝑛▒〖𝑓_𝑖 [(𝑥_𝑖−𝐴)−(𝑥 ̅−𝐴)]^𝑟 〗
𝑚_𝑟=𝑚_𝑟^′−(_^𝑟)𝐶_1 .𝑚_(𝑟−1)^′.𝑚_1^′+(_^𝑟)𝐶_2 .𝑚_(𝑟−2)^′ (𝑚_1^′ )^2− . . .
+(_^𝑟)𝐶_(𝑟−1) .𝑚_1^′.〖(−1)^(𝑟−1).(𝑚_1^′ )〗^(𝑟−1)+(−1)^𝑟.(𝑚_1^′ )^𝑟
𝑚_2=𝑚_2^′−〖 (𝑚_1^′ )〗^2
𝑚_3=𝑚_3^′− 3.𝑚_2^′.𝑚_1^′+ 2.(𝑚_1^′ )^3
𝑚_4=𝑚_4^′− 4.𝑚_3^′.𝑚_1^′+ 6.𝑚_2^′.(𝑚_1^′ )^2− 3. (𝑚_1^′ )^4
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Three types of moments are:
1. Moments about arbitrary point 2. Moments about origin 3. Moments about mean
A Moment about Origin is a special case of a Moment about an arbitrary point where the arbitrary point is set to origin 0.
Moments about Mean (also called Central moments)
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Why Moments are important?
a. Describe the characteristic of a distribution
b. Represent a convenient and unifying method for summarizing several statistical measures such as measures of tendency, variation, skewness and kurtosis.
#moment #deviation #mean #variance #momentrelation #psnacademy
00:00 Expression of rth moment
01:16 Deduction of the relation
05:55 Binomial theorem
02:23 Applying Binomial expansion
19:14 Putting r=2
21:02 Putting r=3
22:31 Putting r=4
24:14 Summary