CS代考程序代写 Statistics 100B

Statistics 100B
University of California, Los Angeles Department of Statistics
Practice problem – week 3
Instructor: Nicolas Christou
Answer the following questions:
a. Suppose the number of pine trees in a certain forest follows the Poisson distribution with parameter λ per meter2. Suppose we randomly select a point (say A) in this forest (not a pine tree, just a point). Let X be the distance from this point to the nearest pine tree and let Y be the distance from this point to the second nearest pine tree (see graph below). Find the probability density function of X and then show that the random variable λπX2 follows the exponential distribution with mean 1. Note: The parameter λ here is given per meter2. The parameter λ of a circle with radius r is λπr2.
d. Lets=λπ
i=1i
(Y −X ). Ifsandtareindependentshowthat i=1i i
􏰃i=1
m ∼beta(m,m).
●
●
x y
A
●
b. Refer to question (a). Find the probability density function of Y . (x is fixed when we are considering the pdf of Y .) Show that the random variable λπ(Y 2 − X2) follows the exponential distribution with mean 1.
c. Suppose now we randomly select m points in this forest. Find the distribution of 2λπ􏰃m Xi2 and the
􏰃m 2 2 distribution of 2λπ i=1(Yi − Xi ).
􏰃m 2
X andt=λπ
􏰃m 2 2
i=1 􏰃m Xi2
1
Y2 i=1 i