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- 2. Let X1, X2, ..., X, be iid random variables with a common pdf/pf f(x, 8) depending on an unknown parameter 0. Using the factorisation criterion show that any one-to-one function of a sufficient statistic is a sufficient statistic.[Bayes' Theorem, Discrete Random Variables] It is known that 90% of new cars produced by a certain company have on average À malfunctions per 1 km of covered distance. Besides this, it is known that remaining 10% of new cars were produced with a defect which increases the average malfunction rate to 2λ. The two types of cars are indistinguishable otherwise. When (a) Suppose that only a single malfunction has occured with a randomly selected car over 1,000 km of distance. What is the probability that the car is from the defective group, and how does it depend on λ? (b) For a randomly selected car, find a minimum distance covered with no malfunctions for the probability that the car is from the non-defective group is 95% or above (express the result as a function of X).For a random variable X, suppose that E[X] = 1 and Var(X) = 5. Then %3D (a) E[(2 + X)²] = (b) Var(2 + 3X) = Netsu rtiol orodit on this problem
- Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.Without using a moment generating function; Prove that the variance of a beta-distributed random variable with parameters α and β is σ2 = αβ/[(α + β)^2 (α + β + 1)]!
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