Question: Let X be a random variable with cdf F(x) and let T (F) be a functional. We say that T (F) is a scale functional

Let X be a random variable with cdf F(x) and let T (F) be a functional. We say that T (F) is a scale functional if it satisfies the three properties

3 (i) T(Fax) (ii) T(Fx+b) (iii) T(F_x) = aT (Fx), T(Fx), T(Fx).

Show that the following functionals are scale functionals.
(a) The standard deviation, T (FX) = (Var(X))^1/2.
(b) The interquartile range, T (FX) = F⁻¹_X (3/4) − F⁻¹_X (1/4).

3 (i) T(Fax) (ii) T(Fx+b) (iii) T(F_x) = aT (Fx), T(Fx), T(Fx). for a > 0 for all b

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