Solution Found!
Show that 0 ? ai ? 1, i = 1, 2, ..., then
Chapter 3, Problem 12TE(choose chapter or problem)
Show that \(0 \leq a_{i} \leq 1, i=1,2, \ldots\), then
\(\sum_{i=1}^{\infty}\left[a_{i} \prod_{j=1}^{i-1}\left(1-a_{j}\right)\right]+\prod_{i=1}^{\infty}\left(1-a_{i}\right)=1\)
Hint: Suppose that an infinite number of coins are to be flipped. Let \(a_i\) be the probability that the ith coin lands on heads, and consider when the first head occurs.
Questions & Answers
QUESTION:
Show that \(0 \leq a_{i} \leq 1, i=1,2, \ldots\), then
\(\sum_{i=1}^{\infty}\left[a_{i} \prod_{j=1}^{i-1}\left(1-a_{j}\right)\right]+\prod_{i=1}^{\infty}\left(1-a_{i}\right)=1\)
Hint: Suppose that an infinite number of coins are to be flipped. Let \(a_i\) be the probability that the ith coin lands on heads, and consider when the first head occurs.
ANSWER:Step 1 of 2
From the given problem we have:
for
We need prove the above relation.