Question

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

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

Step-by-step solution

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.

Explanation (Part a)

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

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

$P(X=i,X+Y=n)=\frac{1}{n-1}$

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

$$\begin{gathered} P(X=i,X+Y=n)=\frac{P(X=i,X+Y=n)}{P(X+Y=n)} \\ =\frac{P(X=i)P(Y=n-i)}{P(X+Y=n)} \\ =\frac{1}{n-1} \end{gathered}$$

Hence proved.