Suppose X1,...,Xn are i.i.d. real-valued random variables with c.d.f. F. Assume 1 < 2 such that F(1)=1/4,

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 inter-quartile range (defined as the difference between the .75 quantile and .25 quantile) is a √n- consistent estimator of the population inter-quartile range (θ2 − θ1).