1. (20%) Assume that random variables X, Y have a joint probability density function (pdf) f(x, y) = 1/8, 0 y 4; y x y + 2 (a) (5%) Compute E[Y|X = x]

2. (30%) Let X₁, X₂, ..., Xₙ be a random sample drawn from the pdf $$f(x) = e^{(-(x-\theta))}, x > \theta; -\infty < \theta < \infty$$ and F(x) denotes the corresponding distribution function. (a) $(5\%)$ Let $Y{1} = \min \{X{1},\dots,X{n}\}$. Show that $Y{n}$ is a consistent estimator of $\theta$.