Let X = (X1, X2)T follow a bivariate normal distribution with mean vector (0, 0)T and covariance matrix Σ such that: Σ = ⌈ 5 -3 ⌉ ⌊ -3 10 ⌋ The mean vector and covariance matrix of Y = (X1, 5 − 2X2)T are:

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?