Let X1, …, X7 and Y1, …, Y9 be two random samples drawn independently from two populations with continuous CDFs F and G respectively. Consider the Wald-Wolfowitz run test in the context of the following two sample testing problems: H0: F(x) = G(x) ∀ x vs. H1: F(x) ≠ G(x) for some x. If the random variable R denotes the total number of runs in the combined ordered arrangement of the two given samples, then which of the following is true?

Let X1, X2, ... be a sequence of independent and identically distributed random variables having a Binomial(3, 14) distribution. For j = 1, 2, ..., define Yj = 1 if Xj ≤ √5, and Yj = 0 otherwise. If 1n ∑j=1n Yj2 converges almost surely to a constant c as n → ∞, then the value of c is:

Let X be a single sample from an absolutely continuous distribution with probability density function f(x|θ) = 2θ2 (θ - x) for 0 0 is unknown. Which of the following intervals is a 95% confidence interval for θ?