Exercise: Estimate the area of the flower whose boundary has the equation *r* = √(2 + cos7*θ*) in polar coordinates.

(Pick points at random in an enclosing rectangle and count the fraction of points that are inside the flower.)

Probability density definition

*Thought exercise* (Quiz): The
random numbers generated by our LCG, and by numpy.random.rand(),
have a uniform density on [0,1]: p(u) = 1.

If we took sums of pairs of such numbers, what would the probabilty density of those sums be? Without the computer, think about it and sketch what you think the histogram of the sums of pairs will look like.

After that, let's check experimentally.

Let's make sums of triples, sums of quadruples etc., and see how the PDF changes.

Can we shift and stretch or shrink the sums so that the PDF tends to something as n goes to infinity?

Radioactive decay and exponential distribution