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Question

# Consider a finite sequence of random values X={x1, x2, x3.....xn} Let μx be the mean and σx be the standard deviation of X. Let another finite sequence Y of equal length be derived from this yi=a.xi+b, where a and b are positive constants. Let μy be the mean and σy be the standard deviation of this sequence. Which one of the following statements is incorrect?

A
Index position of mode of x in X is the same as the index position of mode of y in Y
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B
Index position of median of x in X is the same as the index position of median of y in Y
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C
μy=aμx+b
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D
σy=aσx+b
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Solution

## The correct option is D σy=aσx+b y = ax + b ⇒ var (y) = Var(ax) + Var(b) =a2Var(x)+0 [∵ Var (const) = 0] or σy=a2σx Alternatively: mean, median and mode are linear function over a random variable. So, multiplying by constants or adding constants won't change their relative position. Standard deviation is not a linear function over a random variable. So, option (d) is correct.

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