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Chapter 6: Q.6.18 (page 276)
Suppose X and Y are both integer-valued random variables. Let p(i|j) = P(X = i|Y = j) and q(j|i) = P(Y = j|X = i) Show that P(X = i, Y = j) = p(i|j) i p(i|j) q(j|i
The required expression is
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The joint density of X and Y is
Find the conditional distribution of Y, given X = x.
Suggest a procedure for using Buffon’s needle problem to estimate π. Surprisingly enough, this was once a common method of evaluating π.
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}.
Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute
(a) P{X1 > X2|X1 > X3};
(b) P{X1 > X2|X1 < X3};
(c) P{X1 > X2|X2 > X3};
(d) P{X1 > X2|X2 < X3}
Let X1, X2, ... be a sequence of independent and identically distributed continuous random variables. Find
a)
b)
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