Consider a frequency distribution consisting of the first n natural numbers, where each number i occurs with a frequency equal to i . If the sum of the variance and the square of the mean of this distribution is 105 , then the value of n is equal to

Step-by-step solution

For any frequency distribution, the relationship between variance ^2 and mean is given by ^2 = f_i X_i^2 f_i - ^2 . Thus, ^2 + ^2 = f_i X_i^2 f_i . In this distribution, the variate values are X_i = i and their corresponding frequencies are f_i = i for i = 1, 2, , n . The sum of the frequencies is: f_i = _ i=1 ^ n i = n(n+1) 2 The sum of the squares of the variates multiplied by their frequencies is: f_i X_i^2 = _ i=1 ^ n (i)(i^2) = _ i=1 ^ n i^3 = n^2(n+1)^2 4 Substituting these into the expression for ^2 + ^2 giv