Suppose X1,..., Xn are i.i.d. real-valued random variables with c.d.f. F. Assume 1 < 2 such that

Suppose X1,..., Xn are i.i.d. real-valued random variables with c.d.f. F. Assume ∃θ1 < θ2 such that F(θ1) = 1/4, F(θ2) = 3/4, and F is differentiable, with density f taking positive values at θ1 and θ2. Show that the sample interquartile range (defined as the difference between the 0.75 quantile and 0.25 quantile)

is a √n- consistent estimator of the population inter-quartile range (θ2 − θ1).