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Chapter 6: Q.6.14 (page 275)

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).

Short Answer

Expert verified

X + Y = n, we have that X has discrete uniform distribution over1,......,n-1

Step by step solution

01

Content Introduction

A random variable is a variable that has a set of possible values and probability. It's a variable whose value is determined by the outcome of an unknown event.

02

Explanation (Part a)

In the nth trial we have obtained head for the second time.

Whereases, in first n-1trial we have obtained n-2tails and one head. But to observe the arrangement of n-2tails and only one head, it seems to happen with the same probability. Hence we can conclude

P(X=i,X+Y=n)=1n-1

03

Explanation (Part b)

Random variable X + Y has negative binomial distribution with parameter of success p and we obtain two successes. Thus, we have that

P(X=i,X+Y=n)=P(X=i,X+Y=n)P(X+Y=n)=P(X=i)P(Y=n-i)P(X+Y=n)=1n-1

Hence proved.

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Most popular questions from this chapter

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) 鈭� R 0 otherwise

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(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 f(x,y)=24xy0鈮�x鈮�1,0鈮�y鈮�1,0鈮�x+y鈮�1and let it equal 0 otherwise.

(a) Show that f(x,y)is a joint probability density function.

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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:

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Let X1, X2, ... be a sequence of independent and identically distributed continuous random variables. Find

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