X₁, ..., X₅ are independent and uniform on (0, 1). What is P(max Xᵢ ≤ 0.8)?
A0.8
B0.8⁵ ≈ 0.328
C1 - 0.2⁵ ≈ 0.9997
D5 × 0.8 = 4
The maximum is at most 0.8 exactly when all five values are at most 0.8. By independence the probability is 0.8⁵ ≈ 0.328. The expression 1 - 0.2⁵ is the probability that the minimum is at most 0.8.
Question 2 True / False
The minimum of n independent exponential random variables, each with rate λ, is exponential with rate nλ.
TTrue
FFalse
Answer: True
P(min > x) = P(all Xᵢ > x) = (e^(-λx))ⁿ = e^(-nλx). That is the survival function of an exponential distribution with rate nλ. For example, the first of n independent components to fail does so at rate nλ.
Question 3 Short Answer
For an independent uniform(0, 1) sample of size n, what is the expected value of the k-th order statistic?
Think about your answer, then reveal below.
Model answer: k/(n + 1)
X₍ₖ₎ has a Beta(k, n - k + 1) distribution, whose mean is k/(k + (n - k + 1)) = k/(n + 1). The n points split (0, 1) into n + 1 gaps that have equal expected length 1/(n + 1).
Question 4 Short Answer
Explain where the factor n!/((k - 1)!(n - k)!) in the density of X₍ₖ₎ comes from.
Think about your answer, then reveal below.
Model answer: It counts the ways to choose which k - 1 observations fall below x, which one falls at x, and which n - k fall above x.
For X₍ₖ₎ to be near x, exactly k - 1 values lie below x (probability F(x) each), one lies in a small interval at x (probability about f(x)dx), and n - k lie above (probability 1 - F(x) each). The number of ways to assign the n observations to these three groups is the multinomial coefficient n!/((k - 1)! 1! (n - k)!).