नमूने (sample) 1, 3, 1, 8, 2, 8, 3, 8, 2, 13 के आधार पर एक चरघातांती (exponential) समष्टि के दर पैरामीटर (λ) का अधिकतम संभाव्यता अनुमान (maximum likelihood estimate) क्या है?

Let X1, X2, ..., Xn (n ≥ 2) be a random sample from a continuous distribution with the probability density function: f(x | θ) = (1/(2θ)) e-|x-θ|/θ, x ∈ R where θ (> 0) is an unknown parameter. Let Un = (1/n) ∑i=1n Xi, Vn = (1/n) ∑i=1n Xi2, and Sn2 = (1/(n-1)) ∑i=1n (Xi - Un)2. Then, which of the following statements is true?

Let X1, X2, ..., Xn be independent and identically distributed random variables with the probability density function: f(x | θ) = e-(x-θ) if x ≥ θ, and 0 otherwise where θ ∈ R is the unknown parameter. Let Y = min(X1, X2, ..., Xn). Which of the following is a confidence interval for θ with the confidence coefficient (1-α), where α ∈ (0, 1)?