1. (25 points) The following passage is from the Wikipedia page for the numbers
ID: 2929010 • Letter: 1
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1. (25 points) The following passage is from the Wikipedia page for the numbers game. The numbers game, also known as the numbers racket, the policy racket, the policy game or the daily number is a form of illegal gambling or illegal lottery played mostly in poor and working class neighborhoods in the United States, wherein a bettor attempts to pick three digits to match those that will be randomly drawn the following day. In recent years, the "number" would be the last three digits of "the handle,"the amount race track bettors placed on race day at a major racetrack, published in racing journals and major newspapers in New York. Essentially the game has 1000 equally likely outcomes: the three digit sequences from 000-999. A $1 dollar bet will pay out $600 if you guessed the numbers in the proper sequence. We are going to consider the outcome in terms of the net change in money in your pocket after bet: it's either-1 or 599. The mean for this random variable is -4 -0.40. Joe rnakes a $1 bet ver lay or-many years. Explain what the Law of Large Numbers says about Joey's results as he keeps betting? (See note and figure below) -Tha+- ·the'ns| yous nould braa kever 'n To help you with this, I conducted a simulation of 10,000 daily $1 bets and plotted the running average in the manner of the applet associated with Problem 15.4 in your text. Refer to Figure 1 2. (25 points) Again, for a single bet the mean net change of money in your pocket is =-0.40. The standard deviation for this random variable is 18.96. Say Joe makes 14,000 $1 bets over the course of a very long time. The Central Limit Theorem says that the average net change in his wallet after the 14,000 plays is approximately normally distributed. (a) (10 points) What are the mean and standard deviation of this distribution?Explanation / Answer
1. The law of large numbers say that the probability(in the long run, Joe will have lost -0.4 in all) is tending towards 1.
The problem says that under the betting scheme, Joe either profits $599 or loses $1. So if X is the random variable giving the net earn figures of Joe then given that its mean value is -0.4. So, the large numbers law states that a variable tends towards mean in long run with probability 1.So essentially, If the game continues in this fashion for a very large number of days, then Joe will net have -0.4 dollars as net earn, meaning will have expectedly lost 0.4 dollars.
2. 14,000 days have been played, hence the number of days is very high and we can apply the central limit theorem which states that if X is a random variable with mean m_x and s.d. s_x then [ sqrt(n) {X(bar) - m_x} / s_x ] follows a N(0,1) distribution, where n is the number of trials and X9bar) is the average long run net gain/loss. The mean of X as given in the problem is -0.4 dollars and standard deviation is 18.96 (already provided in the problem).
3. The average net loss or gain is X(bar) which follows a normal distribution with mean m_x = -0.4 and standard deviation s_x = 18.96/sqrt(n). (from (2)). Also the number of trials i.e. n=14,000. So
if Y is the average net loss or gain then Y follows N(-0.4, (0.16)2 ). => (Y+0.4)/0.16 ~ N(0,1)
The problem asks us to calculate P(-0.5 < Y < -0.3) = P( {(-0.5+0.4)/0.16} < {Y+0.4)/0.16} < {(-0.3+0.4)/0.16} )
= P ( -0.63 < Z < 0.63 ) where Z follows N(0,1)
=P(Z<0.63) - P(Z<-0.63) = Phi(0.63) - (1-Phi(0.63) ) = 2*Phi(0.63) - 1 = 0.47 [calculated from Biometrika table]
where Phi(.) is the C.D.F of N(0,1), since Normal is symmetric about 0, hence P(Z<-0.63) = 1 - P(Z<0.63).
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