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ISyE 6644 — Summer 2009
Homework #2 Solutions
- Use the basic Monte Carlo technique from class to integrate
I =
∫ (^1)
− 1
2 π
exp{−x^2 / 2 } dx.
(a) Use n = 100 Unif(0,1) random variates to produce your answer. Repeat this 50 times and make a histogram of the results. (b) Using the above 50 runs, estimate the variance of Iˆ 100. (c) Can you think of any way to calculate an “exact” answer?
By the way, if you need a uniform generator, here’s an easy FORTRAN function that a lot of people use: (Of course, if you don’t want to do any programming, you can use Excel; or you can just pick a whole bunch of uniforms out of your head — but this might get a little tedious!!)
FUNCTION UNIF(IX)
K1 = IX/127773 (Note: this division truncates, e.g., 5/3 = 1.) IX = 16807(IX - K1127773) - K1*2836 (seed is updated for next use) IF(IX.LT.0)IX = IX + 2147483647 UNIF = IX * 4.656612875E- RETURN END
Solution. (a)
Iˆn = b^ −^ a n
∑^ n
i=
f (a + (b − a)Ui)
∑^100
i=
f (−1 + 2Ui)
(since a = −1, b = 1, and n = 100)
=
∑^100
i=
2 π
e−(−1+2Ui)
2 .
Now you do the rest (50 times). 2
(b) Just take the sample variance of your 50 replications from (a). 2
(c) The actual answer is
I =
∫ (^1)
− 1
2 π
exp{−x^2 / 2 } dx
= 2Φ(1) − 1 (where Φ(·) is the standard normal c.d.f.) = 2(0.8413) = 0. 6826. 2
- Let’s make some pi. Consider a unit square with a circle of radius 1/2 inscribed in it. Toss a dart randomly at the square. The probability that the dart will hit inside the circle is the ratio of the area of the circle to that of the square, i.e., π/4. If you toss n darts at the square, the proportion ˆpn that hit the circle converges to π/4. Therefore, the random variable 4ˆpn is an estimator for π. Implement this Monte Carlo scheme in your favorite language to estimate π. Plot your estimates for sample sizes of n = 2, 4 , 8 ,... , 2048. (Use a logarithmic scale for the x-axis.) Does your answer seem to be converging to 3.14159? (I hope so.)
Hints: You may need to recall that the relevant circle is described by (x − 1 /2)^2 + (y − 1 /2)^2 ≤ 1 /4. Generate the ith dart by using [U 1 i, U 2 i], where the U ’s are i.i.d. Unif(0,1) random variables.
Solution. Lots of ways to do this problem. If you follow the hint, simply deter- mine the proportion of time that your darts fall inside the circle, say ˆpn. Then, as discussed in class, the estimator for π is ˆπn = 4ˆpn. You should be getting slow but pretty steady convergence to π as the sample size gets larger. 2
- With the previous question in mind, implement a Monte Carlo scheme to estimate the volume in a sphere with radius 1/2.
Solution. Your answer should converge to 43 πr^3 = π/6. 2
