91Ó°ÊÓ

Let N be a geometric random variable with parameter p. Suppose that the conditional distribution of X given that N = n is the gamma distribution with parameters n and λ. Find the conditional probability mass function of N given that X = x.

An ambulance travels back and forth at a constant speed along a road of length L. At a certain moment of time, an accident occurs at a point uniformly distributed on the road. [That is, the distance of the point from one of the fixed ends of the road is uniformly distributed over (0, L).] Assuming that the ambulance’s location at the moment of the accident is also uniformly distributed, and assuming independence of the variables, compute the distribution of the distance of the ambulance from the accident.

Suppose that X and Y are independent geometric random variables with the same parameter p.

(a) Without any computations, what do you think is the value of P{X = i|X + Y = n}?

Hint: Imagine that you continually flip a coin having probability p of coming up heads. If the second head occurs on the nth flip, what is the probability mass function of the time of the first head?

(b) Verify your conjecture in part (a).

Let X and Y be independent uniform (0, 1) random variables.

(a) Find the joint density of U = X, V = X + Y.

(b) Use the result obtained in part (a) to compute the density function of V

The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is f(x, y) =  c if(x, y) ∈ R 0 otherwise

(a) Show that 1/c = area of region R. Suppose that (X, Y) is uniformly distributed over the square centered at (0, 0) and with sides of length 2

(b) Show that X and Y are independent, with each being distributed uniformly over (−1, 1).

(c) What is the probability that (X, Y) lies in the circle of radius 1 centered at the origin? That is, find P{X² + Y²< 1}.

Suppose that n points are independently chosen at random on the circumference of a circle, and we want the probability that they all lie in some semicircle. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line, as shown in the following diagram:

Let P1, ... ,Pn denote the n points. Let A denote the event that all the points are contained in some semicircle, and let Ai be the event that all the points lie in the semicircle beginning at the point Pi and going clockwise for 180â—¦, i = 1, ... , n.

(a) Express A in terms of the Ai.

(b) Are the Ai mutually exclusive?

(c) Find P(A).

If X and Y are independent binomial random variables with identical parameters n and p, show analytically that the conditional distribution of X given that X + Y = m is the hypergeometric distribution. Also, give a second argument that yields the same result without any computations. Hint: Suppose that 2n coins are flipped. Let X denote the number of heads in the first n flips and Y the number in the second n flips. Argue that given a total of m heads, the number of heads in the first n flips has the same distribution as the number of white balls selected when a sample of size m is chosen from n white and n black balls

Three points $X_1,X_2,X_3$ are selected at random on a line $L$. What is the probability that$X_2$ lies between $X_{1}and{X}_3$?

Three points X₁, X₂, X₃ are selected at random on a line L. What is the probability that X₂ lies between X₁ and X₃?

Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X and Y are independent random variables such that X is uniformly distributed over (0, L/2) and Y is uniformly distributed over (L/2, L).] Find the probability that the distance between the two points is greater than L/3