Consider the one-dimensional Poisson process with intensity λ. Show that the number of points in [0, t], given that the number of points in [0, 2t] is equal to n, has a Bin(n, 1 2 ) distribution....

Consider the one-dimensional Poisson process with intensity λ. Show that the number of points in [0, t], given that the number of points in [0, 2t] is equal to n, has a Bin(n, 1 2 ) distribution. Hint: write the event {N([0, s]) = k, N([0, 2s]) = n} as the intersection of the (independent!) events {N([0, s]) = k} and {N((s, 2s]) = n − k}.