Let x ¯ , M and σ 2 be respectively the mean, mode and variance of n observations x 1 , x 2 , . ... , x n and d i = - x i - a , i = 1 , 2 , . ... , n , where a is any number. Statement I: Variance of d 1 , d 2 , . . . , d n is σ 2 . Statement II: Mean and mode of d 1 , d 2 , . . . . , d n are - x ¯ - a and - M - a , respectively.

Question

Let x ¯ , M and σ 2 be respectively the mean, mode and variance of n observations x 1 , x 2 , . ... , x n and d i = - x i - a , i = 1 , 2 , . ... , n , where a is any number. Statement I: Variance of d 1 , d 2 , . . . , d n is σ 2 . Statement II: Mean and mode of d 1 , d 2 , . . . . , d n are - x ¯ - a and - M - a , respectively., Statement I and Statement II are both true, Statement I and Statement II are both false, Statement I is true and Statement II is false, Statement I is false and Statement II is true

MARKS App · Accepted Answer

Mean of d 1 , d 2 , d 3 . . . d n = ∑ i = 1 n - x i - a n = - ∑ i = 1 n x i n + - a n ∑ i = 1 n 1 = - x + - a n n = - x → - a And, the mode of d 1 , d 2 , d 3 . . . d n will be - M - a Also, we know that the variance σ 2 is unaffected by change of origin, hence it remains same.

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