Think about the (weak) Law of Large Numbers, which says thatthe average of n i.i

Question

Think about the (weak) Law of Large Numbers, which says thatthe
average of n i.i.d random variables, Xi, each with mean andvariance 2, converges in
probability to . For example, if each Xi is Bernoulli(p) fora coin that lands heads with
probability p = , then Xn p in probability.

A man plays a gambling game which is close to fair (i.e., chance hewins any particular
play is a little less than 0.5, and all plays are independent).After losing 8 plays in a row,
he states “By the law of averages (law of large numbers), mychances of winning on each
play should now increase to compensate for my run of bad luck, so Ishould keep
gambling another 5-10 times as I am destined to win.”

Do you agree or disagree with this man? Briefly, why? Your answermust use and
interpret the WLLN in some way.

Explanation / Answer

Yes, we should agree with the man's statement. Explanation: When we toss a fair coin only twice, althogh "Heads" and"Tails" both have a probability of 0.5, you certainly wouldn't betoo surprised if the two tosses happened to both produce "Heads",or both produce "Tails", instead of producing exactly one "Head"and one "Tail". But if we toss the same fair coin 1000 times, you certainlyexpect the number of "Heads" to be very close to 500. In the given example also we are having 50% chances of thehappening of the event. So, this intution can be backed by the Weak Law of LargeNumbers. In words, this Law states that if a trail is reproducedlarge number of times n, then it becomes exceedingly improbablethat the average of the outcomes of these n trials will differsignificantly from the expected value of one outcome as n growswithout limit. Interpreation: No matter how small , all you have to do to make theprobability for the mean of the first n terms to differ from themean by more than to be as small as you wish is tomake n large enough. In the vocabulary of estimation, the WLLN states that thesample mean is the consistent estimator of the populationmean. No matter how small , all you have to do to make theprobability for the mean of the first n terms to differ from themean by more than to be as small as you wish is tomake n large enough. In the vocabulary of estimation, the WLLN states that thesample mean is the consistent estimator of the populationmean.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.