Consider (X1, X2, X3) with the joint pdf f(x1, 12, 13) x exp{-} (x² + x² + x} - {₁x2-x2x3)}. The full conditionals for X₂X₁ = 1, X3 = 23 and X3|X₁ = ₁, X2 = 1₂ for this joint density are X2|X₁ = x1, X3 = 23 ~ N (₁ + x3,7). X3|X1 = 21, X₂ = 2₂ ~ N(2, 1). You do not need to show how to obtain these two conditional distributions. (a) Find the full conditional for X₁|X2 = 22, X3 = 23. (b) Given current values, and the ability to generate random variables from any univariate normal distribution, show how Gibbs sampling can be used to obtain the next set of sample values from the joint distribution. Make sure you explicitly give any univariate normal distributions used along the way and use the notation from the course. (c) It turns out that Gibbs sampling is not actually necessary to simulate from this joint distribution. This is because, marginally, X₁~ N(0,3) and, conditionally, X₂X₁ = ₁ ~ N (₁,3) while, as above, X3|X₁ = 21, X₂ = x2 ~ N(2, 1). Explain carefully how you can generate a sample of values from the joint distribution using this information (and the ability to generate random variables from any univariate normal distribution).