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I was looking at the expected output of an algorithm and translated it into this equivalent problem:

Consider $n$ points sampled uniformly from $[-1,1]^2$ and let $A=\frac{4Z}{n}$ where $Z$ is how many of these $n$ points lie within the interior and boundary of the unit circle. Find $\mathbb{E}[A]$.

I tried running the algorithm but I fail to recognize if the output value is approximating a familiar number. How does one compute for this expected value?

AB Balbuena
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1 Answers1

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  • The probability that a single point lies in the unit circle is $p := \frac{\text{area of unit circle}}{\text{area of $[-1,1]^2$}}$.
  • $E[Z] = np$ (why?)
  • $E[A] = \frac{4}{n} E[Z]$ (why?)
angryavian
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