Let X be a random sample of size 1 from the probability density function: f(x|θ) = 3/θ3(θ - x)2, if 0 < x < θ, and 0 otherwise. If (x/(1 - λ1), x/(1 - λ2)) is a confidence interval for θ with confidence coefficient 1 - α, where λi ∈ (0,1), i = 1,2, λ1 < λ2, and α ∈ (0, 1), then which of the following statements 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?