To calculate the probability that a randomly selected point ((x, y)) lies below the curve (y = x) in the interval [,1], we can use both analytical integration and the Monte Carlo method.

  1. Define the Integral: The problem is to find the probability that (y \leq x) where (x) and (y) are uniformly distributed over the interval [,1]. This translates to finding the area under the curve (y = x) from 0 to 1, divided by the total area of the unit square (which is 1).

  2. Set Up the Integral: [ P = \int{}^{1} x \, dx ] Calculate the integral: [ \int{}^{1} x \, dx = \left[ \frac{x^2}{2} \right]_^{1} = \frac{1^2}{2} - \frac{^2}{2} = \frac{1}{2} ]

  3. Monte Carlo Method:

    • Generate (N = 1) random points within the interval [,1].
    • Check if (y \leq x).
    • The number of points under the curve is 5.
    • The probability estimate is: [ \text{Probability} = \frac{5}{1} = 0.5 ]

Conclusion

Both methods give the same probability of 0.5.

[ \boxed{\dfrac{1}{2}} ]

Analytical Integration 第1张

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