the first four moments of a distribution about the value 2 are 1, 2.5, 5.5 and 16 respectively. the second centeral moment (second moment about actual mean) is
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Given :- The first four moments of a distribution about the value 4 of a variable are -1.5, 17, -30 and 108 . Find the moment about mean. ?
Formula used :-
The first moment around mean is Equal to = {(x1)¹ + (x2)¹ + (x3)¹ + _________ (xn)¹} / n = (x1 + x2 + x3 + ______ xn)/n = (sum of observation) / (Total Number of observation).
Solution :-
value of 4 variables are given as :-
x1 = (-1.5)
x2 = 17
x3 = (-30)
x4 = 108
So,
→ sum of variables = (-1.5) + 17 + (-30) + 108 = (-1.5) + (-13) + 108 = (-14.5) + 108 = 93.5
Therefore,
→ Moment about Mean = (x1 + x2 + x3 + x4) / 4 = (93.5/4) = 23.375 .(Ans.)
Hence, the moment about mean is 23.375.
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Extra :-
The 2nd moment around the mean = {(x1)² + (x2)² + (x3)² + _________ (xn)²} / n = Σ(xi - μx)² = Variance.
The 3rd moment around the mean = {(x1)³ + (x2)³ + (x3)³ + _________ (xn)³} / n = Skewness.
The 4th moment around the mean = {(x1)⁴ + (x2)⁴ + (x3)⁴ + _________ (xn)⁴} / n = Kurtosis.