Some Hints

From Stanford Wong's BJ21

Posted by MathProf on 18 Dec 1997, at 5:03 p.m., in response to Re: Nice post professor ... Derivation?, posted by Chris C on 18 Dec 1997, at 2:40 p.m.

It is easy when you know how to do it... I started working on it once, and got caught up in a huge mess of Binomial Coefficients and all sorts of things. Then I looked in a Stats Book and it was easier.

I'll give you an outline. I will do it for the HiLo Count, which is balanced, which makes it a shade easier.
Suppose we have a sample of n cards. Let X be the RUnning Count fo the sample. We can write X as a sum of X1+X2+...+Xn, where each Xi is the HiLO value of card i. Problem is to determine the Variance of X.

Now Var(X) = Sum [ Var(Xi) ] + Sum Cov(Xi,Xj)

The second sum is taken over all distinct ( i,j ). (or sets of {i,j} if you multiply by 2). The first sum is over i=1..n.

Now the Var(Xi) is 10/13 for the hiLo Count. For the Covariance terms, they are all the same. We are asking if you pull out two different cards, what is the Covarianace of their HiLo Count ... This is 


  Prob(X1=1 and XJ=1) * 1 + Porb(Xi=-1 & Xj=-1) * 1 

   +  Prob(X1=-1 & Xj=1) (-1) + Prob(Xi=1 * Xj=-1 *(-1)

In calcualating these convariances, is where the "without replcaement" comes in.

If you have trouble finishing it, let me know, and I'll try to add some more details.

Where in Griffin did you see the formula?

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