Suppose X ∼ Binomial(n, p). This exercise relates the distribution function of the binomial to the beta
distribution. (a) Show that for p ∈ (0, 1), P[U(k+1) > p] = P[X ≤ k]. [Hint: The (k + 1) st order statistic is greater than p if and only if how many of the Ui 's are less than p? What is that probability?] (b) Conclude that if F is the distribution function of X, then F(k) = P[Beta(k + 1, n − k) > p]. This formula is used in the R function pbinom.
The first part is straightforward. The second part can be solved using the following formula, if U 1... View the full answer