 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/8/b/iii/solution

For

n \geq 2

X_{2} + \dots + X_{n}

is independent of

X_{1}

(n - 1) λ

. Part (ii) therefore gives

X_{1} ∣ (X_{1} + \dots + X_{n} = n) \sim Bin (n, 1/ n) .

Its probability at zero is

P (X_{1} = 0 ∣ X_{1} + \dots + X_{n} = n) = (1 - \frac{1}{n})^{n} (n \geq 1) .

For

n = 1

the event being conditioned on is

X_{1} = 1

, so the answer is

0

, also matching the displayed expression. The cancellation of

λ

reflects the equal rates of all

n

counts, not an approximation or large-

n

limit.

 New to topics? Read the docs here!