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Let X N (0, 1) and Y N (0, 1) be independent random

Fundamentals of Probability, with Stochastic Processes | 3rd Edition | ISBN: 9780131453401 | Authors: Saeed Ghahramani ISBN: 9780131453401 223

Solution for problem 3 Chapter 8.4

Fundamentals of Probability, with Stochastic Processes | 3rd Edition

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Fundamentals of Probability, with Stochastic Processes | 3rd Edition | ISBN: 9780131453401 | Authors: Saeed Ghahramani

Fundamentals of Probability, with Stochastic Processes | 3rd Edition

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Problem 3

Let X N (0, 1) and Y N (0, 1) be independent random variables. Find the joint probability density function of R = X2 + Y 2 and 5 = arctan(Y/X). Show that R and 5 are independent. Note that (R, 5) is the polar coordinate representation of (X, Y ).

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Exam 4: April 11, 2016 1 The Russian Revolutions of 1917  Conditions/problems on the Eastern Front. o The lack of weapons was the major problem for the Russian Army.  1 in 3 soldiers did not have a rifle.  The Russians had ¼ the artillery of the Germans.  Pounded by the German army.  2 million Russian soldiers were killed.  4­6 million Russian soldiers were wounded or captured.  The February Revolution. o 10,000 women in Petrograd held a demonstration. o Shops and factories closed down. o Transportation had stopped running.  What Is To Be Done o The revolution in Russia will be led by a small party of professional revolutionaries.  Lenin’s campaign for supporters. o “Peace, Land, Bread, Now”. o Workers will have control of industry.  Leon Trotsky and the Red Guard. o The revolution in Russia will be led by a small party of professional revolutionaries. o Led the October Revolution. o Training for the siege of ower.  The October Revolution. o The Bolshevik seizure of power. o Took over communication centers in Petrograd.  Important developments after the October Revolution. o Peace treaty signed with Germany. o Civil War 1918­1921. o Lenin’s death and Stalin’s rise to power.  Fate of Nicholas II and his family. o Execution.  Lenin’s “Testament”. o Wanted someone else in his place besides Stalin. o Not endorsing Stalin as his successor. o Was not made public until 1958.  Stalin’s falsification of photographs. o Made a lot of “fake” photos to achieve his end. o Became a trend in altering photographs to alter records. Stalin’s “Revolution from Above”  Five Year Plans. o Wanted rapid industrialization in a short amount of time. o High goals to project targets.  Conditions of everyday life for industrial workers. o Food rationing. o Consumer goods were in short supply.  Clothes.  Shoes.  Home goods. o Very little emphasis placed on consumer goods.  Stalin’s industrial development: the view from below. Exam 4: April 11, 2016 2 o Loaded the working class with socialist competition, shock work, over­fulfilling the industrial and financial plan, and so forth. o Toils seven hours not leaving his post. o Sits in endless meetings or classes for 1 ½­ 2 hours in order to increase his skill level.  Collectivization of agriculture. o Transform from an agricultural for a greater output for the land.  Feed industrial workers.  Export grain in order to obtain equipment. o Order peasants.  Give up their private land and their livestock.  Relocated. o Collective farms.  Transformed private agriculture into small state­run collective farms.  People were set on farms to meet quotas.  Complete in most areas of the Soviet Union.  Kulaks. o Peasants who resisted collectivization. o Burnt their fields and killed their livestock in order to resist the Soviet Union.  Famine 1932­3. o Claimed about 3­5 million lives. o Let nature run its course and let people die.  The Great Terror. o Campaign to purge people who were enemy of the people.  Essential to the success of the country.  People from all rules of life were considered and enemy. o Nearly 1 million people killed.  1.5 million people sent to labor camps.  Propaganda—A.O. Advienko. o Everything he is and everything he has is because of Stalin. o Stalin is the new God in the Soviet Union. o Be willing to work hard and sacrifice for Stalin. o Paintings.  The “Cut of Stalin”.  Stakhanovites. o Make people work harder for their goods. o “Heroes of labor”. The Nazi State  Hitler’s economic programs. o Enabling Act.  Granted Hitler ultimate power for 4 years.  4 year plan turned into a 12 year dictatorship. o Public work projects.  Batted unemployment.  Built up construction.  Autobahm.  Engineers and workers would find jobs easily.  Hitler Youth. o Get the ready for their future...

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Chapter 8.4, Problem 3 is Solved
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Textbook: Fundamentals of Probability, with Stochastic Processes
Edition: 3
Author: Saeed Ghahramani
ISBN: 9780131453401

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Let X N (0, 1) and Y N (0, 1) be independent random