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Download Rab Ne Bana Di Jodi (Dancing Jodi) *HD* 1080p *Blu Ray*… Video Song Mp3  .Q:

When a mean value over $N$ angles, that change from moment to moment, converges?

The problem is:
For the function $f(x, y) = xy^2$, we have a vector field $mathbf{F} = (-x^2 y, -y^2 x)$ on the unit square. For $N$ angles on the unit square taken at random from $theta_1, dots, theta_N$, we compute $N$ averages, $bar{f}_N (theta_i) = frac{1}{N} sum_{i=1}^N f(x_i, y_i)$. Let $F_N (theta) = lim_{N to infty} bar{f}_N (theta)$. Show that $F_N (theta)$ is a linear function of $theta$.
The first question I have is: why do we take the limit as $N$ goes to infinity? Doesn’t that assumption (the number of samples is large) violate the original problem statement?
The second question is that I am having trouble understanding how we would find the mean value of $F_N (theta)$. Should we look at the slope of the function? I would greatly appreciate it if anyone could provide a link or detail how to solve this problem!
Thanks in advance.

A:

This comes from an application of the law of large numbers.
Anytime you have a random variable $theta_i$ and a sequence of functions ${f_n}$ and you apply the weak law of large numbers to the sequence ${theta_i}_1^infty$ and the function $f$, you get a random variable $F_theta$ such that $$E[F_theta]=lim_n
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