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2022 Spring. Solutions of M262 Final Exam problems. 1. Aslı constructed a machine producing plastic balls with average weight µ and standard deviation σ. Knowing these two parameters, she states that the probability that the total weight of 81 balls produced by this machine is less than 180 is equal to 0.44 and the probability that the total weight of 81 balls produced by this machine is greater than 90 is equal to 0.33. Based on this information find µ and σ. Solution 1: P(x1 + · ...

Solutions of QUIZ 5. M262 Section 1. (25.04.2022). 1. A zoologist is going to test a Null Hypothesis that the standard deviation σ of the weights of frogs in some region is equal to 5 grams. She collected 11 frogs, measured their weights and found a sample standard deviation s = 3. Based on this data should Null Hypothesis σ = 5 be rejected in favor of Alternative Hypothesis σ 6= 5? Use α = 0.1. Solution: H0 : σ = 5 H1 : p 6= 5 For d = 10 χ 2 (0.95) = 3.94 and χ 2 (0.95) = 18...

2022 Spring. Solutions of M262 Midterm problems. 1a. 16 balls are randomly distributed to 7 empty boxes. Given that one of the boxes contains 9 balls find the probability that there is no empty box. Solution: Let A be the even that one of the boxes contains 9 balls and B be the event that no box is empty. Then P(A) =  16 9 7 1  6 7 7 16 (we choose a box containing 9 balls, choose 9 balls which go to this box and distribute remaining 7 balls. P(B ∩ A) =  16 9 7 2 ...

Solutions of QUIZ 6. M262 Section 1. (27.04.2022). 1. A study is to be made of the relative effectiveness of two kinds of cough medicines in increasing sleep. Seven people with colds are given medicine A the first night and medicine B the second night. Their hours of sleep each night are recorded. The hours of sleep of 7 people in the first night are 4.9, 4.2, 5.9, 5.0, 5.2, 7,5, 6.3, respectively. The hours of sleep of 7 people in the second night are 4.0, 4.3, 5.1, 5.0, 5.3, 7,2, 6.0, r...

1. An automobile manufacturer is interested in determining the relationship between the size and manufacturer of newly purchased automobiles. One thousand recent buyers of American-made cars are randomly sampled, and each purchase is classified with respect to the size and manufacturer of the automobile. Out of 341 cars of manufacturer A 157 are small, 126 are intermediate, 58 are large. Out of 192 ca

Solution of QUIZ 2. M262 Section 1. (21.02.2022). 1. A box contains four coins: two fair coins, a two-headed coin and a coin which shows up heads with probability 1/11. We randomly select one of the coins and toss it. Given that it shows up heads find the probability that it is not a fair coin. Solution: By Bayes’ formula P = 1/4 · 1 + 1/4 · 1/11 1/4 · 1/2 + 1/4 · 1/2 + 1/4 · 1 + 1/4 · 1/11 = 12 23 . 2. A box contains 2 white and 3 red balls. We draw balls from the box one-by...

EEE 448/548 - Reinforcement Learning & Dynamic Programming Solutions to Final Exam Problem 1. (30pt) Consider the following infinite-horizon Markov decision process with the discount factor γ = 1 and initialized at state s1: At each step, the agent stays in state s1 and receives reward 1 if he/she takes action a1, and receives reward 0 and terminates the process otherwise. We focus on (Markov) stationary policy parametrized by a single parameter θ as follows πθ(a1 | s1) = θ and πθ(...

Solutions of QUIZ 4. M262 Section 1. (04.04.2022). 1. From interviews with 1000 adults it was found that 780 adults supported tougher legislation for antipollution measures. Does this poll substantiate the conjecture that more than 75% of the adult population are in favor of tougher legislation for antipollution measures? Use α = 0.1. Calculate the P-value. Solution: H0 : p = p0 H1 : p > p0 where p0 = 0.75.

IE—456/556 & EEE-448/548 %% Reinforcement Learning and Dynamic Programming Final Exam - Summer 2022 Duration: 150 minutes Name Surname: Bilkent ID: Signature: Q1: Pacman Bonus Level! o o o (€] 1 2 3 4 o) Pacman is in a bonus level! With no ghosts around, he can eat as many dots as he wants. He is in the 5 x 1 grid shown above, where the cells are numbered from left to right, that is, s € {1,...,5}. In cells 1 through 4, the actions available are to move Right (R) or t...

MATH 255 - Probability and Statistics Final Exam Solutions 5 January 2025 Problem 1. [10pt] The joint pdf of random variables X and Y is given by: fX,Y (x, y) =    c if (x, y) ∈ S, 0 otherwise, where c is a constant and S is the set shown in the plot. (a) Find the least mean square (LMS) estimator g(X) of Y . E[Y |X] =    0.5 if X ∈ [0, 1] ∪ [2, 3] , 1 if X ∈ [1, 2]. Before we start solving the problem, let us first find the value of c. Since the joint...