**2.6. Exercises for Chapter 2**

*Exercise 2.1.*

We are looking at a circular target *D* of center 0 and of radius *R* which is used for archery. The couple *Z* = (*X*, *Y*) represents the coordinates of the point of impact of the arrow on the target support; we assume that the r.v. *X* and *Y* are independent and following the same law *N* (0.4*R*^{2}).

1) What is the probability that the arrow reach the target?

2) How many times must one fire the arrow In order for, with a probability ≥ 0.9, the target is reached at least once (we give *n* 10 ≠ 2.305).

Let us assume that we fire 100 times at the target, calculate the probability that the target to reached at least 20 times.

Hint: use the central limit theorem.

*Solution 2.1.*

1) The r.v.s *X* and *Y* being independent, the density of probability of *Z* = (*X, Y*) is *f*_{Z} (*x, y*) = *f*_{X} (*x*) *f*_{Y} (*y*) = and *P* (*Z ∈ D*) = using a change from Cartesian to polar coordinates:

2) At each shot *k*, we associate a Bernoulli r.v. *U*_{k} ∼ *b* (*p*) defined by

In *n* shots, the number of impacts is given ...