STAT 3128 HW # 4 Solutions Spring 2013 Instr. Sonin

Transcription

1 STAT 28 HW 4 Solutions Spring 2 Instr. Sonin Due Wednesday, March 2 NAME ( points) Show all work on problems! (5). p. [5] 58. Please read subsection 6.6, pp Change:...select components...to: select and... accept the batch only if...at most 4. (correction) in c) change;...with replacing 2... to: with replacing 4; in d) change;...with 5 replacing... to: with 25 replacing 2; Probability ( batch is accepted) as a function of the operating characteristic curve a) 25, of defective components; 25.; ;.25 ) 4 using the table e.; 5 ;.25 a) again using the table e c).; 5 ; d 2, of defective components; 2 ) table d.; 5 ; e) Which of the three sampling plans appears more satisfactory? To answer this question we need to know in fact the cost of false acceptance and the cost of false rejection, but generally we want a plan for which accept is high for and low for. The plan in c) seems better in this regard.

2 (4) 2. A test for the presence of a certain disease has probability. 4 of giving a false-positive reading (a positive test for a healthy person) and probability. of giving a false-negative result. Suppose that three individuals are tested, two of whom do not have the disease and one of whom has. Let the number of negative readings. a) Does have a binomial distribution? Explain your reasoning. No b/c though these three trials are indep. but the pr-ty of S is not the same in each trial b) What is the probability that at least two of the three test results are negative? Let us denote events {no disease}, { disease}, test is negative Success. Then ) 96 ), 7 Let say first two are healthy and the last one is a carrier Then, where is Bin 2 ) 96 is Bin ) Then 2 Let Then 2 2! 7!$ 97 c) Find the expected number of negative tests. % % % % (4). Alice and Bob begin to play a sequence of chess games. Let wins a game, and suppose that outcomes of successive games are independent with &.6 and.4 (they never draw). They will play until one of them wins two games in a row or three games in total. Let the number of games played. Find the probability distribution (pmf) of and the expected value of

3 5 ( ( ( ( ( ) ( ( * ( ( ( ( ( for 6( 4 ; & & ( & & & ( ( ( ( ( 4 & & & & ( ( 5 & & & & & & & & & & ) ( ( * ( ( ( % , (6) 4. The weekly demand for...is a rv with pdf + if otherwise. (a) Calculate cdf of,,.5,, 5, ; if - - a) cdf& if if & + hence.,, & ,, 5 & &! 75 5; & $ (b) Calculate % /.2;

4 % % = To find median, we have to solve an eq-n & Its approximate solution is $ 2 & 2$ 5 (c) If 5 units are in stock at the beginning of the week, how much is expected to be left at the end of the week? the remaining amount if 5 if or 5 25 ; & 5+ 5!75! $ (6) 5. A system consists of three components and is working if at least two components are working. Suppose that -th component has a lifetime that is exponentially distributed with, 2 2 and and that components fail independently. Let the time at which the system fails. a) Calculate the cdf &, the pdf of and the expected value of. Hint: use notation lifetime of -th component,, and consider events, and use the fact that % 6 7 for 6 % b) Determine the probability that the system will not fail before 6 hours. Sol-n: Let 4 of working (after components. We have

5 4 and pdf 2!!!! Now is NOT an exp rv (! ) but we can use the fact that for an exp rv with parameter ( ) 2 Therefore % 45.! b) Determine the probability that the system will not fail before 6 hours $ $ 25 ( 5)*!. The Bernoulli-Laplace model, is a simple discrete model for the diffusion of two incompressible gases (fliuids) between two containers. It can be formulated as a simple ball and urn model. Thus, suppose that we have two urns, labeled and 2. Each contains 9 balls, Urn contains red balls and 9 blue balls and urn 2 contains + red and 9 + blue balls. At each discrete time, independently of the past, a ball is selected at random from each urn and then the two balls are switched. The balls of different colors correspond to molecules of different types, and the urns are the containers. The incompressible property is reflected in the fact that the number of balls in each urn remains constant over time. For 9 + find the limit distribution in both containers. Solution: This is a simple Markov Chain. There are three possible states of such system: { 2:; where 2 < < : < < and ; < < The probability of transition from state 2 to state: 2: the probability of transition from state ; to state: 2; and :2 to select from urn and select < from urn 2) similarly : c to select < from urn and select from urn 2) and :: Such system tends to an equlibrium. Let - and = be the probabilities of these states in equlibrium. These probbailities must satisfy the balance equations: probability in probability out - = = and of course - = The unique solution is > $