What is the probability density of the time between the arrival of the two packets of Ex...

What is the probability density of the time between the arrival of the two packets of Example E in Section 3.4?

Reference

Suppose that a node in a communications network has the property that if two packets of information arrive within time τ of each other, they “collide” and then have to be retransmitted. If the times of arrival of the two packets are independent and uniform on [0, T ], what is the probability that they collide? The times of arrival of two packets, T₁ and T₂, are independent and uniform on [0, T ], so their joint density is the product of the marginals, or

for t₁ and t₂ in the square with sides [0, T ]. Therefore, (T₁, T₂) is uniformly distributed over the square. The probability that the two packets collide is proportional to the area of the shaded strip in Figure 3.12. Each of the unshaded triangles of the figure has area (T −τ)²/2, and thus the area of the shaded area is T ²−(T −τ)². Integrating f (t₁, t₂) over this area gives the desired probability: 1 − (1 − τ/T )².