# Question

Suggest a procedure for using Buffon’s needle problem to estimate π. Surprisingly enough, this was once a common method of evaluating π.

## Answer to relevant Questions

Let X(1) ≤ X(2) ≤ . . . ≤ X(n) be the ordered values of n independent uniform (0, 1) random variables. Prove that for 1 ≤ k ≤ n + 1, P{X(k) − X(k−1) > t} = (1 − t)n where X(0) ≡ 0, X(n+1) ≡ t. (a) If X has a gamma distribution with parameters (t, λ), what is the distribution of cX, c > 0? (b) Show that has a gamma distribution with parameters n, λ when n is a positive integer and χ22n is a chi-squared random ...Let A1, A2, . . . ,An be arbitrary events, and define Ck = {at least k of the Ai occur}. Show that Let X denote the number of the Ai that occur. Show that both sides of the preceding equation are equal to E[X]. A coin having probability p of landing on heads is flipped n times. Compute the expected number of runs of heads of size 1, of size 2, and of size k, 1 ≤ k ≤ n. How many times would you expect to roll a fair die before all 6 sides appeared at least once?Post your question

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