Beautiful homework # 4 ENGR 323 CESSNA Page 1/5

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1 Beautiful homework # 4 ENGR 33 CESSNA Page 1/5 Problem 3-14 An operator records the time to complete a mechanical assembly to the nearest second with the following results. seconds number of assemblies Table 1 (Top row is seconds bottom row is number of assemblies completed) Recall definition 3-1, p.105 The function f ()=P(X=) from the set of possible values of the discrete random variable X to the interval [0,1] is called a probability mass function. For a random variable X, f () satisfies the following properties: (1) F () = P(X=) () F () > or = 0 for all (3) Σ f () = f() 0.15 u = V(X)=-4. V(X)= Figure 1 (Probability Mass Function for problem 3-14) a) Determine the probability mass function of X. In other words what will the percentage of completed assemblies be at each (number of seconds) of the random variable X? Divide the number of completed assemblies associated with a given time by the total number of completed assemblies. f (30) = 3/1 f (31) = 5/ (number of seconds)

2 Beautiful homework # 4 ENGR 33 CESSNA Page /5 f (3) = 6/1 f (33) = 9/1 f (34) = 1/1 f (35) = 5/1 f (36) = 3/1 f (37) = 15/1 f (38) = 9/1 f (39) = 6/1 Note that the sum of these probabilities is 1 as required by definition 3-1, p.105. b) Determine P(33 <38) In other words just add up the percentages of assemblies for each that is equal to 33 seconds and on up to but less than 38 seconds. f (33 <38) = f (33)+f (34)+f (35)+f (36)+f (37) = ( )/9 = 93/1 c) What proportions of the assemblies is completed in 35 or less seconds? This is similar to part b, where we just added up percentages. f ( 35) = f (30)+f (31)+f (3)+f (33)+f (34)+f (35) = ( )/1 = 60/1 Problem 3-7 Continuation of eercise 3-14 Determine the cumulative distribution function CDF for the random variable in eercise 3-14; and also determine the following probabilities. Recall definition 3-, p.109 and the 3 CDF properties The cumulative distribution function of a random discrete variable X, denoted as F (), is F () = P(X ) = f( i ) i a) P(X<3.5) = 8/1 b) P(X 3) = 14/1 c) P(X>3) = 1- P(X 3) = 14/1

3 Beautiful homework # 4 ENGR 33 CESSNA Page 3/5 d) P(33< 38) = f (38)-f (33) = (116/1) (3/1) = 93/1 F() (seconds) Figure (Cumulative Distribution Function for problem 3-14) Problem 3-39 Continuation of eercise 3-14 Determine the mean and variance of the random variable in a) Determine the mean of the random variable X. Recall definition 3-3, p.113 The mean or epected value of the random variable X U = E(X) = f () U = E(X) = 30(3/1)+31(5/1)+ +39(6/1) = 35.4 To see U refer to figure 1. b) Determine the variance of the random variable X. Recall definition 3-4, p.116 The variance V of a random variable X V(X) = E(X-U) = ( ) U f ()

4 Beautiful homework # 4 ENGR 33 CESSNA Page 4/5 V(X) = ( ) (3/1)+ +( ) ( 6 /1) To see V(X) refer to figure 1. = 4.5 Problem 3-15 Continuation of eercise Suppose that the times of two assemblies are recorded. Determine the range of each of the random variables below. Recall definition p.10. A discrete random variable is a random variable with a finite (or countably infinite) range. In these problems the the number of seconds and the number of completed assemblies are easily countable, we will not need to epress the probabilities of the random variable X into a formula. Refer to table one for the following four questions. a) Total time of the two assemblies They re asking for the range of times depending on any two assemblies choosen, for eample (30,30) would equal sity, (30,31) would equal sity-one. Let Y be a random variable representing any 1 +, then the whole range of Y is (60,61,6,,77,78) b) Average time of the two assemblies This time they just want the range of averages of the random variable Y. So lets let Z denote this Z = Y/ = ( 1 + )/,then the range of these values is (30, 30.5, 31,,38.5, 39) C) Difference in time of the two assemblies Now they re asking for the range of differences for any two assemblies, so for eample if we picked (30,30) the difference is zero, (30,39) the difference is nine or negative nine depending which way one subtracted. Let D be a random variable and denote the range of differences for all possibilities D = (-9,-8 7,,0,, 7, 8, 9) D) The longest time of the two assemblies In other words if two assemblies are selected at random, which of the two had taken the longest time to complete. For eample if ( 1, ) are chosen and unless they had both taken the same amount of time to complete, then one of them has taken longer than the other. Let X be the random variable of time, then the range of possible times would be (30, 31,, 38, 39)

5 Beautiful homework # 4 ENGR 33 CESSNA Page 5/5

Math 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.

Math 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential. Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample