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Advanced Computer Graphics: Difference between revisions
→Monte Carlo Integration
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==Monte Carlo Integration== | ==Monte Carlo Integration== | ||
Suppose we want to estimate <math>I_1 = \int_{a}^{b}f(x)dx</math>.<br> | |||
Then we can use <math>\hat{I_1} = \frac{b-a}{N}\sum f(X_i)</math> where <math>X_1,...,X_n \sim Uniform(a,b)</math>.<br> | |||
This is because <math>E[\frac{b-a}{N}\sum f(X_i)] = \frac{b-a}{N}\sum E[f(X_i)] = \frac{1}{N}\sum \int_{a}^{b}(b-a)f(x)(1/(b-a))dx = \int_{a}^{b}f(x)dx</math><br> | |||
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