Consider the unit real line segment 0 1 two points X1 and X2

Consider the unit real line segment [0, 1]; two points X1 and X2 are picked uniformly at random where the (random) variable represents the distance from the origin. Denote the minimum and maximum as follows:
X(1)=min{X1,X2} X(2)=max {X1,X2}
(i) Write an expression for the c.d.f. for the minimum, i.e. FX(1)(t)=P { X(1) t}
(ii) What is probability that both points chosen are to the right of the point 0 < t < 1 ?
- Solve the above problem in two ways as suggested –
(a) Sketch the event set in R2 and assign the probability;
(b) Use the result in (i) directly.

Solution

(i)

X₁~U(0,1) and X₂~U(0,1) and both of them are independent

X₍₁₎=min(X1,X2)

The cdf of X(1) is given by,

F_X(1)(t)=P[ X₍₁₎ t ]=1-P[X₍₁₎>t]=1-P(X₁>t,X₂>t)=1-(P(X₁>t))²=1-(1-t)²=2*t-t² ,0<t<1

(since X₁ and X₂ are indeoendent random samples)

(ii)

P[Both points chosen are to the right of the point t]=P[X₍₁₎>t]=1-F_X(1)(t)=1-2*t+t² =(1-t)²