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RR The following sample program computes numerically double integral U cos10x + 20y dxdy over a circle x2 + y2 = 1. The only conceptual di culty as compared to single integrals is how to select points at random from the unit disk. This is done by picking points from a bigger square and discarding those that didn't make it. Can you do this integral analytically? Or by another numerical procedure? bas ' 'declarations DECLARE FUNCTION Integrand!  DECLARE FUNCTION InDomain! 4. 96 * SQRVar N - Sum END FUNCTION InDomain X, Y 'This function checks if $x,y$ is in the integration domain 'The definition of the domain can be easily modified here, including 'more complicated domains IF X ^ 2 + Y ^ 2 1 THEN InDomain = True END FUNCTION FUNCTION Integrand X, Y 'This is the function to be integrated Integrand = COS10 * X + 20 * Y ' END FUNCTION It is important to have some idea about how accurate the answer is.

3 on page 5 assuming that the arrival times are exponential rather than uniform. Assume independence. 28 There are 3 stop-lights spaced within 1km of each other and operating asynchronously.  Assuming each is red for 1 minute and then green for one minute, what is the average time to pass through the three lights by a car that can instantaneously accelerate to 60km h. This exercise can be developed into a simulation project that may address some of the following questions How does the speed change with the number of lights?

Are determined by the cumulative distribution function F x = PrX  x 1:12 The corresponding tail function Rx = 1 , F x = PrX x is sometimes called the reliability11 function. Cumulative distribution function can be used to express probabilities of intervals Pra X  b = F b , F a. a+ Pra X  b = F a+ , F a. The right hand side limit F a+ exists, as F is a non-decreasing function. 9 Suppose F x = 1 , e,x _ 0. Then PrjX , 2j 1 = e,1 , e,3. In probability theory we are concerned with probabilities.

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