If statements and For loops
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1 Advanced Random Simulation If statements and For loops In order to incorporate more complex aspects into the model, use If statements and For loops. If[condition,t,f]
2 Advanced Random Simulation If statements and For loops In order to incorporate more complex aspects into the model, use If statements and For loops. If[condition,t,f] First, Mathematica evaluates the condition. If condition is true, the statement evaluates to t. If condition is false, the statement evaluates to f.
3 Advanced Random Simulation If statements and For loops In order to incorporate more complex aspects into the model, use If statements and For loops. If[condition,t,f] First, Mathematica evaluates the condition. If condition is true, the statement evaluates to t. If condition is false, the statement evaluates to f. Examples of conditions: x<0 (x==0) && (y!=1) RandomInteger[]==1 Note the double equals sign == and not equals!=.
4 Advanced Random Simulation If statements and For loops In order to incorporate more complex aspects into the model, use If statements and For loops. If[condition,t,f] First, Mathematica evaluates the condition. If condition is true, the statement evaluates to t. If condition is false, the statement evaluates to f. Examples of conditions: x<0 (x==0) && (y!=1) RandomInteger[]==1 Note the double equals sign == and not equals!=. Examples. If[x<0, -x, x] is the absolute value function. Why?
5 Advanced Random Simulation If statements and For loops In order to incorporate more complex aspects into the model, use If statements and For loops. If[condition,t,f] First, Mathematica evaluates the condition. If condition is true, the statement evaluates to t. If condition is false, the statement evaluates to f. Examples of conditions: x<0 (x==0) && (y!=1) RandomInteger[]==1 Note the double equals sign == and not equals!=. Examples. If[x<0, -x, x] is the absolute value function. Why? If[RandomInteger[] == 1, "Head", "Tail"] gives Head half the time and gives Tail half the time.
6 Advanced Random Simulation Using If statements in Table commands Goal: Model a 7.5% chance of occurrence. Recall that RandomReal[] outputs a random number between 0 and 1. To model a 7.5% chance of occurrence, split the interval at. 0 1
7 Advanced Random Simulation Using If statements in Table commands Goal: Model a 7.5% chance of occurrence. Recall that RandomReal[] outputs a random number between 0 and 1. To model a 7.5% chance of occurrence, split the interval at Anything to the left of the split will be taken as success.
8 Advanced Random Simulation Using If statements in Table commands Goal: Model a 7.5% chance of occurrence. Recall that RandomReal[] outputs a random number between 0 and 1. To model a 7.5% chance of occurrence, split the interval at Anything to the left of the split will be taken as success. To model this is Mathematica, use an If statement. trial = RandomReal[] success = If[trial <= 0.075, 1, 0]
9 Advanced Random Simulation Using If statements in Table commands Goal: Model a 7.5% chance of occurrence. Recall that RandomReal[] outputs a random number between 0 and 1. To model a 7.5% chance of occurrence, split the interval at Anything to the left of the split will be taken as success. To model this is Mathematica, use an If statement. trial = RandomReal[] success = If[trial <= 0.075, 1, 0] Alternatively, do this is one step: If[RandomReal[] <= 0.075, 1, 0]
10 Advanced Random Simulation Using If statements in Table commands That was: If[RandomReal[] <= 0.075, 1, 0] Let s run this command many times and visualize the results: Remember that Table will repeat a command multiple times: trials=table[if[randomreal[] <= 0.075, 1, 0],{500}];
11 Advanced Random Simulation Using If statements in Table commands That was: If[RandomReal[] <= 0.075, 1, 0] Let s run this command many times and visualize the results: Remember that Table will repeat a command multiple times: trials=table[if[randomreal[] <= 0.075, 1, 0],{500}]; Output: 500-entry list, where each entry is 0 (failure) or 1 (success).
12 Advanced Random Simulation Using If statements in Table commands That was: If[RandomReal[] <= 0.075, 1, 0] Let s run this command many times and visualize the results: Remember that Table will repeat a command multiple times: trials=table[if[randomreal[] <= 0.075, 1, 0],{500}]; Output: 500-entry list, where each entry is 0 (failure) or 1 (success). Question: How many successes? (Expected value: = 37.5)
13 Advanced Random Simulation Using If statements in Table commands That was: If[RandomReal[] <= 0.075, 1, 0] Let s run this command many times and visualize the results: Remember that Table will repeat a command multiple times: trials=table[if[randomreal[] <= 0.075, 1, 0],{500}]; Output: 500-entry list, where each entry is 0 (failure) or 1 (success). Question: How many successes? (Expected value: = 37.5) If we add the entries Total[trials], we get # successes. One time I ran it had 32 successes.
14 Advanced Random Simulation Using If statements in Table commands That was: If[RandomReal[] <= 0.075, 1, 0] Let s run this command many times and visualize the results: Remember that Table will repeat a command multiple times: trials=table[if[randomreal[] <= 0.075, 1, 0],{500}]; Output: 500-entry list, where each entry is 0 (failure) or 1 (success). Question: How many successes? (Expected value: = 37.5) If we add the entries Total[trials], we get # successes. One time I ran it had 32 successes. Alternatively, Tally[trials] gives how many times distinct entries appear. Output: {{0, 468}, {1, 32}}
15 Advanced Random Simulation Using If statements in Table commands That was: If[RandomReal[] <= 0.075, 1, 0] Let s run this command many times and visualize the results: Remember that Table will repeat a command multiple times: trials=table[if[randomreal[] <= 0.075, 1, 0],{500}]; Output: 500-entry list, where each entry is 0 (failure) or 1 (success). Question: How many successes? (Expected value: = 37.5) If we add the entries Total[trials], we get # successes. One time I ran it had 32 successes. Alternatively, Tally[trials] gives how many times distinct entries appear. Output: {{0, 468}, {1, 32}} Last, we might want a visualization; Use Histogram[trials] to get:
16 Advanced Random Simulation If statements and For loops For[start,test,incr,body]
17 Advanced Random Simulation If statements and For loops For[start,test,incr,body] First, Mathematica evaluates the code in start. As long as test is true, (Can happen many times!) Continue to evaluate body and do the increment incr.
18 Advanced Random Simulation If statements and For loops For[start,test,incr,body] First, Mathematica evaluates the code in start. As long as test is true, (Can happen many times!) Continue to evaluate body and do the increment incr. Example. For[i = 0, i < 4, i++, Print[i]] First, Mathematica defines i to be equal to 0. Next, it checks to see if i is less than 4. It is, so it evaluates Print[i], and increments i by 1 (i++).
19 Advanced Random Simulation If statements and For loops For[start,test,incr,body] First, Mathematica evaluates the code in start. As long as test is true, (Can happen many times!) Continue to evaluate body and do the increment incr. Example. For[i = 0, i < 4, i++, Print[i]] First, Mathematica defines i to be equal to 0. Next, it checks to see if i is less than 4. It is, so it evaluates Print[i], and increments i by 1 (i++). Now i = 1, which is still < 4. So Print[i] is evaluated and i is incremented. Similarly for i = 2 and i = 3. Now i is incremented to 4, which is NOT < 4, and the loop terminates.
20 Advanced Random Simulation If statements and For loops For[start,test,incr,body] First, Mathematica evaluates the code in start. As long as test is true, (Can happen many times!) Continue to evaluate body and do the increment incr. Example. For[i = 0, i < 4, i++, Print[i]] First, Mathematica defines i to be equal to 0. Next, it checks to see if i is less than 4. It is, so it evaluates Print[i], and increments i by 1 (i++). Now i = 1, which is still < 4. So Print[i] is evaluated and i is incremented. Similarly for i = 2 and i = 3. Now i is incremented to 4, which is NOT < 4, and the loop terminates. This variable i is called a counter. Be careful to name counters wisely! They are defined as variables.
21 Advanced Random Simulation Simulating flipping a coin Example. Simulate flipping a fair coin 20 times using a for loop. We ll write some pseudocode words that explain what we want the computer to do, but won t actually work if we typed them in.
22 Advanced Random Simulation Simulating flipping a coin Example. Simulate flipping a fair coin 20 times using a for loop. We ll write some pseudocode words that explain what we want the computer to do, but won t actually work if we typed them in. Run the loop 20 times. (Keep track using a counter: let loopcount vary from 1 to 20.) Each time the loop evaluates, Generate a random integer between 0 and 1. If 1 output Head, if 0, output Tail.
23 Advanced Random Simulation Simulating flipping a coin Example. Simulate flipping a fair coin 20 times using a for loop. We ll write some pseudocode words that explain what we want the computer to do, but won t actually work if we typed them in. Run the loop 20 times. (Keep track using a counter: let loopcount vary from 1 to 20.) Each time the loop evaluates, Generate a random integer between 0 and 1. If 1 output Head, if 0, output Tail. For[loopCount = 1, loopcount <= 20, loopcount++, flip = RandomInteger[]; If[flip == 1, Print["Head"], Print["Tail"]]]
24 Advanced Random Simulation Simulating flipping a coin Example. Simulate flipping a fair coin 20 times using a for loop. We ll write some pseudocode words that explain what we want the computer to do, but won t actually work if we typed them in. Run the loop 20 times. (Keep track using a counter: let loopcount vary from 1 to 20.) Each time the loop evaluates, Generate a random integer between 0 and 1. If 1 output Head, if 0, output Tail. For[loopCount = 1, loopcount <= 20, loopcount++, flip = RandomInteger[]; If[flip == 1, Print["Head"], Print["Tail"]]] Notice the == and also the ; that separates the commands.
25 Advanced Random Simulation Simulating flipping a coin Example. Simulate flipping a fair coin 20 times using a for loop. We ll write some pseudocode words that explain what we want the computer to do, but won t actually work if we typed them in. Run the loop 20 times. (Keep track using a counter: let loopcount vary from 1 to 20.) Each time the loop evaluates, Generate a random integer between 0 and 1. If 1 output Head, if 0, output Tail. For[loopCount = 1, loopcount <= 20, loopcount++, flip = RandomInteger[]; If[flip == 1, Print["Head"], Print["Tail"]]] Notice the == and also the ; that separates the commands. loopcount is ONLY a counter; it does not change each step s evaluation.
26 Advanced Random Simulation Simulating flipping a coin Pimp my code! Let s keep track of how many heads and tails are thrown by introducing new counters. headcount will keep track of the number of heads and tailcount will keep track of the number of tails.
27 Advanced Random Simulation Simulating flipping a coin Pimp my code! Let s keep track of how many heads and tails are thrown by introducing new counters. headcount will keep track of the number of heads and tailcount will keep track of the number of tails. Zero out the counters: headcount=0 and tailcount=0. Run the loop 20 times by having loopcount vary from 1 to 20. Each time the loop evaluates, Generate a random integer between 0 and 1. If 1, output Head AND If 0, output Tail AND
28 Advanced Random Simulation Simulating flipping a coin Pimp my code! Let s keep track of how many heads and tails are thrown by introducing new counters. headcount will keep track of the number of heads and tailcount will keep track of the number of tails. Zero out the counters: headcount=0 and tailcount=0. Run the loop 20 times by having loopcount vary from 1 to 20. Each time the loop evaluates, Generate a random integer between 0 and 1. If 1, output Head AND increase headcount, If 0, output Tail AND increase tailcount.
29 Advanced Random Simulation Simulating flipping a coin Pimp my code! Let s keep track of how many heads and tails are thrown by introducing new counters. headcount will keep track of the number of heads and tailcount will keep track of the number of tails. Zero out the counters: headcount=0 and tailcount=0. Run the loop 20 times by having loopcount vary from 1 to 20. Each time the loop evaluates, Generate a random integer between 0 and 1. If 1, output Head AND increase headcount, If 0, output Tail AND increase tailcount. After 20 iterations, display headcount and tailcount.
30 Advanced Random Simulation Simulating flipping a coin Pimp my code! Let s keep track of how many heads and tails are thrown by introducing new counters. headcount will keep track of the number of heads and tailcount will keep track of the number of tails. Zero out the counters: headcount=0 and tailcount=0. Run the loop 20 times by having loopcount vary from 1 to 20. Each time the loop evaluates, Generate a random integer between 0 and 1. If 1, output Head AND increase headcount, If 0, output Tail AND increase tailcount. After 20 iterations, display headcount and tailcount. headcount=0; tailcount=0; For[loopCount = 1, loopcount <= 20, loopcount++, If[RandomInteger[]==1, Print["Head"]; headcount++, Notice the ; Print["Tail"]; tailcount++]] Notice the ++ {headcount, tailcount}
31 Advanced Random Simulation Simulating rolling a biased die Suppose you have a four-sided die, where the four sides (A, B, C, and D) come up with probabilities 1/2, 1/4, 1/8, and 1/8, respectively. 0 1
32 Advanced Random Simulation Simulating rolling a biased die Suppose you have a four-sided die, where the four sides (A, B, C, and D) come up with probabilities 1/2, 1/4, 1/8, and 1/8, respectively. A B C D
33 Advanced Random Simulation Simulating rolling a biased die Suppose you have a four-sided die, where the four sides (A, B, C, and D) come up with probabilities 1/2, 1/4, 1/8, and 1/8, respectively. A B C D Reset the counters: acount=bcount=ccount=dcount=0. For loopcount from 1 to 20, Generate a random real number between 0 and 1. If between 0 and 1/2, then output A and acount++ if between 1/2 and 3/4, then output B and bcount++ if between 3/4 and 7/8, then output C and ccount++ if between 7/8 and 1, then output D and dcount++ Display acount, bcount, ccount, and dcount.
34 Advanced Random Simulation Simulating rolling a biased die acount = 0; bcount = 0; ccount = 0; dcount = 0; For[loopCount = 1, loopcount <= 20, loopcount++, roll=randomreal[]; If[ 0 <= roll < 1/2, Print["a"]; acount++]; If[1/2 <= roll < 3/4, Print["b"]; bcount++]; If[3/4 <= roll < 7/8, Print["c"]; ccount++]; If[7/8 <= roll <= 1, Print["d"]; dcount++];] distribution = {acount, bcount, ccount, dcount}
35 Advanced Random Simulation Simulating rolling a biased die acount = 0; bcount = 0; ccount = 0; dcount = 0; For[loopCount = 1, loopcount <= 20, loopcount++, roll=randomreal[]; If[ 0 <= roll < 1/2, Print["a"]; acount++]; If[1/2 <= roll < 3/4, Print["b"]; bcount++]; If[3/4 <= roll < 7/8, Print["c"]; ccount++]; If[7/8 <= roll <= 1, Print["d"]; dcount++];] distribution = {acount, bcount, ccount, dcount} Sample output: (each on its own line) a, a, a, d, d, b, a, a, d, a, a, a, a, d, b, a, a, c, a, b {12, 3, 1, 4}
36 Advanced Random Simulation Simulating rolling a biased die acount = 0; bcount = 0; ccount = 0; dcount = 0; For[loopCount = 1, loopcount <= 20, loopcount++, roll=randomreal[]; If[ 0 <= roll < 1/2, Print["a"]; acount++]; If[1/2 <= roll < 3/4, Print["b"]; bcount++]; If[3/4 <= roll < 7/8, Print["c"]; ccount++]; If[7/8 <= roll <= 1, Print["d"]; dcount++];] distribution = {acount, bcount, ccount, dcount} Sample output: (each on its own line) a, a, a, d, d, b, a, a, d, a, a, a, a, d, b, a, a, c, a, b {12, 3, 1, 4} These If statements all have no False part. (; vs,)
37 Advanced Random Simulation Simulating rolling a biased die acount = 0; bcount = 0; ccount = 0; dcount = 0; For[loopCount = 1, loopcount <= 20, loopcount++, roll=randomreal[]; If[ 0 <= roll < 1/2, Print["a"]; acount++]; If[1/2 <= roll < 3/4, Print["b"]; bcount++]; If[3/4 <= roll < 7/8, Print["c"]; ccount++]; If[7/8 <= roll <= 1, Print["d"]; dcount++];] distribution = {acount, bcount, ccount, dcount} Sample output: (each on its own line) a, a, a, d, d, b, a, a, d, a, a, a, a, d, b, a, a, c, a, b {12, 3, 1, 4} These If statements all have no False part. (; vs,) Important: You MUST set a variable for the roll. Otherwise, calling RandomInteger four times will have you comparing different random numbers in each If statement.
38 Advanced Random Simulation Simulating rolling a biased die acount = 0; bcount = 0; ccount = 0; dcount = 0; For[loopCount = 1, loopcount <= 20, loopcount++, roll=randomreal[]; If[ 0 <= roll < 1/2, Print["a"]; acount++]; If[1/2 <= roll < 3/4, Print["b"]; bcount++]; If[3/4 <= roll < 7/8, Print["c"]; ccount++]; If[7/8 <= roll <= 1, Print["d"]; dcount++];] distribution = {acount, bcount, ccount, dcount} Sample output: (each on its own line) a, a, a, d, d, b, a, a, d, a, a, a, a, d, b, a, a, c, a, b {12, 3, 1, 4} These If statements all have no False part. (; vs,) Important: You MUST set a variable for the roll. Otherwise, calling RandomInteger four times will have you comparing different random numbers in each If statement. If you are feeling fancy, you can use one Which command instead of four If commands.
39 Advanced Random Simulation Using Simulation to Calculate Area Suppose you have a region whose area you don t know. You can approximate the area using a Monte Carlo simulation. Idea: Surround the region by a rectangle. Randomly chosen points in the rectangle will fall in the region with probability (area of region)/(area of rectangle) We can approximate this probability by calculating (points falling in region)/(total points chosen).
40 Advanced Random Simulation Using Simulation to Calculate Area Example. What is the area under the curve sin(x) from 0 to π? Randomly select 100 points from the rectangle [0,π] [0,1]. [Choose a random real between 0 and π for the x-coordinate and a random real between 0 and 1 for the y-coordinate...] Then, Area of region Number of points in region. 100
41 Advanced Random Simulation Using Simulation to Calculate Area Example. What is the area under the curve sin(x) from 0 to π? Randomly select 100 points from the rectangle [0,π] [0,1]. [Choose a random real between 0 and π for the x-coordinate and a random real between 0 and 1 for the y-coordinate...] Then, Area of region Number of points in region. 100 Here, 63 points fell in the region; we estimate the area to be. Compare this to the actual value, x=π x=0 sin(x)dx = [ cos(x)]x=π x=0 = 2
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