Consider the differential equation rye) – Gry + 12y = 0. Show that the set { is linearly independent.

Consider randomly selecting n segments of pipe and determining the corrosion loss (mm) in the wall thickness for each one. Denote these corrosion losses by Y₁' Yn. The article "A Probabilistic Model for a Gas Explosion Due to Leakages in the Grey Cast Iron Gas Mains"+ proposes a linear corrosion model: Y; = t;R, where t; is the age of the pipe and R, the corrosion rate, is exponentially distributed with parameter 1. Obtain the maximum likelihood estimator of the exponential parameter (the resulting mle appears in the cited article). [Hint: If c> 0 and X has an exponential distribution, so does cX.] O Â = O Â = O λ = * * O n Srit) i = 1 n j = 1 Σ i = 1 n n i = 1 n LY i = 1 n 0 1 = L (rt) 2 i = 1 | = 1 n n Y; | = 1

Calculate the residual of P = (1.2, 2.4) with respect to the line 3x + 4y = 12.

Let {Xi}ni be IID from Bernoulli(p). Find the method of moments estimator of p and use simulations to check whether your estimator is biased or not