JDEP 384H: Numerical Methods in Business

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1 BT 3.4: Solving Nonliner Systems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods Instructor: Thoms Shores Deprtment of Mthemtics Lecture 20, Februry 29, Kufmnn Center Instructor: Thoms Shores Deprtment of Mthemtics

2 BT 3.4: Solving Nonliner Systems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods Outline 1 BT 3.4: Solving Nonliner Systems Univrite Problems Multivrite Problems 2 Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods BT 4.1: Numericl Integrtion BT 4.2: Monte Crlo Integrtion Instructor: Thoms Shores Deprtment of Mthemtics

3 BT 3.4: Solving Nonliner Systems Univrite Problems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Multivrite Methods Problems Outline 1 BT 3.4: Solving Nonliner Systems Univrite Problems Multivrite Problems 2 Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods BT 4.1: Numericl Integrtion BT 4.2: Monte Crlo Integrtion Instructor: Thoms Shores Deprtment of Mthemtics

4 Exmple Exmple A Europen cll option hs strike price $54 on stock with current price $50, expires in ve months, nd the risk-free rte is 7%. Its current price is $2.85. Wht is the implied voltility? Solution. We hve stndrd formul for this sitution tht is stored in the function bseurcll. Get help on it nd use bseurcll to set up n nonymous function of σ using which equls zero when the correct σ is used.

5 BT 3.4: Solving Nonliner Systems Univrite Problems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Multivrite Methods Problems Nonliner Optimiztion for Univrite Functions Bsic Problem: Given function f (x), nd rel number x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (if it exists)? We could solve the eqution f (x) = 0 using ides of root nding bove. Why does this help? Mtlb hs built-in commnd fminbnd tht does not use derivtive informtion, but brcketing procedure. Use Mtlb to minimize f (x) = x 2 sin (x) on intervl [0, 3]. > help fminbnd > fminbnd(myfcn,0,3) > [x,y,exitflg,output]=fminbnd(@(x) x-2*sin(x),0,3) > x = 0:.01:3; > plot(x, x-2*sin(x)) Instructor: Thoms Shores Deprtment of Mthemtics

6 BT 3.4: Solving Nonliner Systems Univrite Problems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Multivrite Methods Problems Nonliner Optimiztion for Univrite Functions Bsic Problem: Given function f (x), nd rel number x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (if it exists)? We could solve the eqution f (x) = 0 using ides of root nding bove. Why does this help? Mtlb hs built-in commnd fminbnd tht does not use derivtive informtion, but brcketing procedure. Use Mtlb to minimize f (x) = x 2 sin (x) on intervl [0, 3]. > help fminbnd > fminbnd(myfcn,0,3) > [x,y,exitflg,output]=fminbnd(@(x) x-2*sin(x),0,3) > x = 0:.01:3; > plot(x, x-2*sin(x)) Instructor: Thoms Shores Deprtment of Mthemtics

7 BT 3.4: Solving Nonliner Systems Univrite Problems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Multivrite Methods Problems Nonliner Optimiztion for Univrite Functions Bsic Problem: Given function f (x), nd rel number x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (if it exists)? We could solve the eqution f (x) = 0 using ides of root nding bove. Why does this help? Mtlb hs built-in commnd fminbnd tht does not use derivtive informtion, but brcketing procedure. Use Mtlb to minimize f (x) = x 2 sin (x) on intervl [0, 3]. > help fminbnd > fminbnd(myfcn,0,3) > [x,y,exitflg,output]=fminbnd(@(x) x-2*sin(x),0,3) > x = 0:.01:3; > plot(x, x-2*sin(x)) Instructor: Thoms Shores Deprtment of Mthemtics

8 BT 3.4: Solving Nonliner Systems Univrite Problems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Multivrite Methods Problems Nonliner Optimiztion for Univrite Functions Bsic Problem: Given function f (x), nd rel number x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (if it exists)? We could solve the eqution f (x) = 0 using ides of root nding bove. Why does this help? Mtlb hs built-in commnd fminbnd tht does not use derivtive informtion, but brcketing procedure. Use Mtlb to minimize f (x) = x 2 sin (x) on intervl [0, 3]. > help fminbnd > fminbnd(myfcn,0,3) > [x,y,exitflg,output]=fminbnd(@(x) x-2*sin(x),0,3) > x = 0:.01:3; > plot(x, x-2*sin(x)) Instructor: Thoms Shores Deprtment of Mthemtics

9 BT 3.4: Solving Nonliner Systems Univrite Problems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Multivrite Methods Problems Outline 1 BT 3.4: Solving Nonliner Systems Univrite Problems Multivrite Problems 2 Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods BT 4.1: Numericl Integrtion BT 4.2: Monte Crlo Integrtion Instructor: Thoms Shores Deprtment of Mthemtics

10 Multivrite Nonliner Optimiztion Bsic Problem: Given sclr vlued function f (x 1, x 2,..., x n ) = f (x), nd vector x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (given tht there is one)? Mny techniques exist (ll of Chpter 6!) Although this is not ecient, theoreticlly one cn turn every root nding problem into n optimiztion problem: to solve the vector eqution f (x) = 0 for x, simply nd the x tht minimizes the sclr function g (x) = f (x) 2. If g (x ) = 0, then f (x ) = 0. So optimiztion is more generl problem thn rootnding. Mtlb does provide multivrite solver clled fminserch. Get help on it nd use it to solve the exmple on the next slide. Try dierent strting points.

11 Multivrite Nonliner Optimiztion Bsic Problem: Given sclr vlued function f (x 1, x 2,..., x n ) = f (x), nd vector x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (given tht there is one)? Mny techniques exist (ll of Chpter 6!) Although this is not ecient, theoreticlly one cn turn every root nding problem into n optimiztion problem: to solve the vector eqution f (x) = 0 for x, simply nd the x tht minimizes the sclr function g (x) = f (x) 2. If g (x ) = 0, then f (x ) = 0. So optimiztion is more generl problem thn rootnding. Mtlb does provide multivrite solver clled fminserch. Get help on it nd use it to solve the exmple on the next slide. Try dierent strting points.

12 Multivrite Nonliner Optimiztion Bsic Problem: Given sclr vlued function f (x 1, x 2,..., x n ) = f (x), nd vector x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (given tht there is one)? Mny techniques exist (ll of Chpter 6!) Although this is not ecient, theoreticlly one cn turn every root nding problem into n optimiztion problem: to solve the vector eqution f (x) = 0 for x, simply nd the x tht minimizes the sclr function g (x) = f (x) 2. If g (x ) = 0, then f (x ) = 0. So optimiztion is more generl problem thn rootnding. Mtlb does provide multivrite solver clled fminserch. Get help on it nd use it to solve the exmple on the next slide. Try dierent strting points.

13 Multivrite Nonliner Optimiztion Bsic Problem: Given sclr vlued function f (x 1, x 2,..., x n ) = f (x), nd vector x with f (x ) = min f (x) over rnge of x vlues. How do we nd solution (given tht there is one)? Mny techniques exist (ll of Chpter 6!) Although this is not ecient, theoreticlly one cn turn every root nding problem into n optimiztion problem: to solve the vector eqution f (x) = 0 for x, simply nd the x tht minimizes the sclr function g (x) = f (x) 2. If g (x ) = 0, then f (x ) = 0. So optimiztion is more generl problem thn rootnding. Mtlb does provide multivrite solver clled fminserch. Get help on it nd use it to solve the exmple on the next slide. Try dierent strting points.

14 Exmple Clcultions Strt by writing out wht the function f : R 2 R 2 dened below ctully represents. > f [x(1)^2-10*x(1) + x(2)^3 + 8; > x(1)*x(2)^2 + x(1) - 10*x(2) + 8] > g norm(f(x))^2 > help fminserch > fminserch(g,[0;0]) > f(ns) > fminserch(g,[2;3]) > f(ns)

15 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Outline 1 BT 3.4: Solving Nonliner Systems Univrite Problems Multivrite Problems 2 Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods BT 4.1: Numericl Integrtion BT 4.2: Monte Crlo Integrtion Instructor: Thoms Shores Deprtment of Mthemtics

16 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b f (x) dx pproximtely when nlyticl methods fil us. Divide intervl [, b] into N equl subintervls by nodes x 0, x 1,..., x N nd width dx = (b ) /N: Left Riemnn sums: I dx Right Riemnn sums: I dx Trpezoidl: I dx 2 N 1 j=0 f (x 0) + f (x j ) N f (x j ). Averge left/right: j=1 N 1 j=1 f (x j ) + f (x N ) Instructor: Thoms Shores Deprtment of Mthemtics

17 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b f (x) dx pproximtely when nlyticl methods fil us. Divide intervl [, b] into N equl subintervls by nodes x 0, x 1,..., x N nd width dx = (b ) /N: Left Riemnn sums: I dx Right Riemnn sums: I dx Trpezoidl: I dx 2 N 1 j=0 f (x 0) + f (x j ) N f (x j ). Averge left/right: j=1 N 1 j=1 f (x j ) + f (x N ) Instructor: Thoms Shores Deprtment of Mthemtics

18 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b f (x) dx pproximtely when nlyticl methods fil us. Divide intervl [, b] into N equl subintervls by nodes x 0, x 1,..., x N nd width dx = (b ) /N: Left Riemnn sums: I dx Right Riemnn sums: I dx Trpezoidl: I dx 2 N 1 j=0 f (x 0) + f (x j ) N f (x j ). Averge left/right: j=1 N 1 j=1 f (x j ) + f (x N ) Instructor: Thoms Shores Deprtment of Mthemtics

19 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b f (x) dx pproximtely when nlyticl methods fil us. Divide intervl [, b] into N equl subintervls by nodes x 0, x 1,..., x N nd width dx = (b ) /N: Left Riemnn sums: I dx Right Riemnn sums: I dx Trpezoidl: I dx 2 N 1 j=0 f (x 0) + f (x j ) N f (x j ). Averge left/right: j=1 N 1 j=1 f (x j ) + f (x N ) Instructor: Thoms Shores Deprtment of Mthemtics

20 Another Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b w(x)f (x) dx pproximtely when nlyticl methods fil us. Here w (x) is nonnegtive weight function nd either or b could be innite. N Motivting formul: I w j f (x j ), where x 1,..., x N re j=1 certin nodes on xed reference intervl nd w 1,..., w N re weights, both of which re computed for once nd for ll. Any other integrl cn be mpped to the reference intervl by simple chnge of vribles. 1 A clssicl exmple (Gussin qudrture): I = 1 f (x) dx 1 Another clssic (Guss-Hermite qudrture, text, p. 216): I = e x2 f (x) dx.

21 Another Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b w(x)f (x) dx pproximtely when nlyticl methods fil us. Here w (x) is nonnegtive weight function nd either or b could be innite. N Motivting formul: I w j f (x j ), where x 1,..., x N re j=1 certin nodes on xed reference intervl nd w 1,..., w N re weights, both of which re computed for once nd for ll. Any other integrl cn be mpped to the reference intervl by simple chnge of vribles. 1 A clssicl exmple (Gussin qudrture): I = 1 f (x) dx 1 Another clssic (Guss-Hermite qudrture, text, p. 216): I = e x2 f (x) dx.

22 Another Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b w(x)f (x) dx pproximtely when nlyticl methods fil us. Here w (x) is nonnegtive weight function nd either or b could be innite. N Motivting formul: I w j f (x j ), where x 1,..., x N re j=1 certin nodes on xed reference intervl nd w 1,..., w N re weights, both of which re computed for once nd for ll. Any other integrl cn be mpped to the reference intervl by simple chnge of vribles. 1 A clssicl exmple (Gussin qudrture): I = 1 f (x) dx 1 Another clssic (Guss-Hermite qudrture, text, p. 216): I = e x2 f (x) dx.

23 Another Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b w(x)f (x) dx pproximtely when nlyticl methods fil us. Here w (x) is nonnegtive weight function nd either or b could be innite. N Motivting formul: I w j f (x j ), where x 1,..., x N re j=1 certin nodes on xed reference intervl nd w 1,..., w N re weights, both of which re computed for once nd for ll. Any other integrl cn be mpped to the reference intervl by simple chnge of vribles. 1 A clssicl exmple (Gussin qudrture): I = 1 f (x) dx 1 Another clssic (Guss-Hermite qudrture, text, p. 216): I = e x2 f (x) dx.

24 Another Numericl Integrtion Bsic Problem: To clculte the denite integrl I = b w(x)f (x) dx pproximtely when nlyticl methods fil us. Here w (x) is nonnegtive weight function nd either or b could be innite. N Motivting formul: I w j f (x j ), where x 1,..., x N re j=1 certin nodes on xed reference intervl nd w 1,..., w N re weights, both of which re computed for once nd for ll. Any other integrl cn be mpped to the reference intervl by simple chnge of vribles. 1 A clssicl exmple (Gussin qudrture): I = 1 f (x) dx 1 Another clssic (Guss-Hermite qudrture, text, p. 216): I = e x2 f (x) dx.

25 Exmples Mtlb uses n dptive Simpson rule, which involves estimting the function s qudrtic over two subintervls, nd using error estimtes to determine if the current pproximtion is good enough. If not, subintervls re further subdivided. > f chis_pdf(x,8) > formt long > N = 40 > dx = (4-0)/N > x = linspce(0,4,n+1); > y = f(x); > Itrue = chis_cdf(4,8) > IMtlb = qud(f,0,4) > Irl = dx*sum(f(x(1:n))) > Irr = dx*sum(f(x(2:n+1))) > Itrp = 0.5*(Irl+Irr) > edit GussInt % don't chnge, just look under the hood > IGqud = GussInt(f,[0,4],3) % try more nodes, up to 8

26 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Outline 1 BT 3.4: Solving Nonliner Systems Univrite Problems Multivrite Problems 2 Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods BT 4.1: Numericl Integrtion BT 4.2: Monte Crlo Integrtion Instructor: Thoms Shores Deprtment of Mthemtics

27 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion The Bsic Ide Monte Crlo Simultion: Crete quntittive model of process. Tret the events tht constitute the process s rndom. Generte rndom vribles to simulte the events. Use these vlues to compute the outcome of the process. A Guiding Exmple is Monte Crlo Integrtion: We wnt to pproximte b g (x) dx. For convenience, ssume g (x) 0, so tht this integrl represents (positive) re. Let's use 1 0 ex dx = e s test cse. Instructor: Thoms Shores Deprtment of Mthemtics

28 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion The Bsic Ide Monte Crlo Simultion: Crete quntittive model of process. Tret the events tht constitute the process s rndom. Generte rndom vribles to simulte the events. Use these vlues to compute the outcome of the process. A Guiding Exmple is Monte Crlo Integrtion: We wnt to pproximte b g (x) dx. For convenience, ssume g (x) 0, so tht this integrl represents (positive) re. Let's use 1 0 ex dx = e s test cse. Instructor: Thoms Shores Deprtment of Mthemtics

29 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion The Bsic Ide Monte Crlo Simultion: Crete quntittive model of process. Tret the events tht constitute the process s rndom. Generte rndom vribles to simulte the events. Use these vlues to compute the outcome of the process. A Guiding Exmple is Monte Crlo Integrtion: We wnt to pproximte b g (x) dx. For convenience, ssume g (x) 0, so tht this integrl represents (positive) re. Let's use 1 0 ex dx = e s test cse. Instructor: Thoms Shores Deprtment of Mthemtics

30 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion The Bsic Ide Monte Crlo Simultion: Crete quntittive model of process. Tret the events tht constitute the process s rndom. Generte rndom vribles to simulte the events. Use these vlues to compute the outcome of the process. A Guiding Exmple is Monte Crlo Integrtion: We wnt to pproximte b g (x) dx. For convenience, ssume g (x) 0, so tht this integrl represents (positive) re. Let's use 1 0 ex dx = e s test cse. Instructor: Thoms Shores Deprtment of Mthemtics

31 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion The Bsic Ide Monte Crlo Simultion: Crete quntittive model of process. Tret the events tht constitute the process s rndom. Generte rndom vribles to simulte the events. Use these vlues to compute the outcome of the process. A Guiding Exmple is Monte Crlo Integrtion: We wnt to pproximte b g (x) dx. For convenience, ssume g (x) 0, so tht this integrl represents (positive) re. Let's use 1 0 ex dx = e s test cse. Instructor: Thoms Shores Deprtment of Mthemtics

32 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion The Bsic Ide Monte Crlo Simultion: Crete quntittive model of process. Tret the events tht constitute the process s rndom. Generte rndom vribles to simulte the events. Use these vlues to compute the outcome of the process. A Guiding Exmple is Monte Crlo Integrtion: We wnt to pproximte b g (x) dx. For convenience, ssume g (x) 0, so tht this integrl represents (positive) re. Let's use 1 0 ex dx = e s test cse. Instructor: Thoms Shores Deprtment of Mthemtics

33 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Monte Crlo Integrtion Hit or Miss Monte Crlo Method: Enclose the grph in box of known re A (in our test cse, 0 x 1, 0 y 3, so A = 3.) Throw N rndom drts t the re, uniformly distributed in x nd y directions. Note: the event of drt throw is represented by rndom pir (X i, Y i ) of independent r.v.'s. Count up the number N H of drts tht fll in the re, i.e., for which Y i g (X i ). Proportiontely, b b g (x) dx A g (x) dx N H N A. N H, so we hve N Instructor: Thoms Shores Deprtment of Mthemtics

34 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Monte Crlo Integrtion Hit or Miss Monte Crlo Method: Enclose the grph in box of known re A (in our test cse, 0 x 1, 0 y 3, so A = 3.) Throw N rndom drts t the re, uniformly distributed in x nd y directions. Note: the event of drt throw is represented by rndom pir (X i, Y i ) of independent r.v.'s. Count up the number N H of drts tht fll in the re, i.e., for which Y i g (X i ). Proportiontely, b b g (x) dx A g (x) dx N H N A. N H, so we hve N Instructor: Thoms Shores Deprtment of Mthemtics

35 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Monte Crlo Integrtion Hit or Miss Monte Crlo Method: Enclose the grph in box of known re A (in our test cse, 0 x 1, 0 y 3, so A = 3.) Throw N rndom drts t the re, uniformly distributed in x nd y directions. Note: the event of drt throw is represented by rndom pir (X i, Y i ) of independent r.v.'s. Count up the number N H of drts tht fll in the re, i.e., for which Y i g (X i ). Proportiontely, b b g (x) dx A g (x) dx N H N A. N H, so we hve N Instructor: Thoms Shores Deprtment of Mthemtics

36 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Monte Crlo Integrtion Hit or Miss Monte Crlo Method: Enclose the grph in box of known re A (in our test cse, 0 x 1, 0 y 3, so A = 3.) Throw N rndom drts t the re, uniformly distributed in x nd y directions. Note: the event of drt throw is represented by rndom pir (X i, Y i ) of independent r.v.'s. Count up the number N H of drts tht fll in the re, i.e., for which Y i g (X i ). Proportiontely, b b g (x) dx A g (x) dx N H N A. N H, so we hve N Instructor: Thoms Shores Deprtment of Mthemtics

37 BT 3.4: Solving Nonliner Systems BT 4.1: Numericl Integrtion Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo BTMethods 4.2: Monte Crlo Integrtion Monte Crlo Integrtion Hit or Miss Monte Crlo Method: Enclose the grph in box of known re A (in our test cse, 0 x 1, 0 y 3, so A = 3.) Throw N rndom drts t the re, uniformly distributed in x nd y directions. Note: the event of drt throw is represented by rndom pir (X i, Y i ) of independent r.v.'s. Count up the number N H of drts tht fll in the re, i.e., for which Y i g (X i ). Proportiontely, b b g (x) dx A g (x) dx N H N A. N H, so we hve N Instructor: Thoms Shores Deprtment of Mthemtics

38 Exmple Clcultion Crry out the following steps in Mtlb > help rnd > formt > rnd('seed',0) > A = 3 > N = 10 > X = rnd(n,1); > Y = (3-0)*rnd(N,1); > hits = sum(y <= exp(x)) > re = A*(hits/N) > Itrue = exp(1)-1 % now try to improve ccurcy

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