Math 255 Introduction to Probability and Statistics

Bilkent University Spring 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 28 May 2021 • Write your name, student number, and Math 255 section number on all your answer sheets. Number the sheets. • The exam consists of three problems. 30 points. • This is a closed book exam. One A4-size two-sided page of notes is allowed. No calculators. • You may receive no credit on correct answers if not fully justified. Simplify sums and integrals where possible to receive full credit. • Good Luck. This study source was downloaded by from CourseH on :51:17 GMT -05:00 1. [10 pts] Let X be a Bernoulli(θ) random variable where θ is modeled as a sample of a random variable Θ uniform on [0, 1]. In other words, we use a Bayesian model for the pair (Θ, X) so that fΘ(θ) = ( 1, 0 ≤ θ ≤ 1, 0, otherwise, and pX|Θ(x|θ) = ( θ, x = 1, 1 − θ, x = 0. (a) (5 pts) Compute the Least Mean Squares (LMS) estimate of Θ, namely Θ( ˆ x) = E[Θ|X = x],