Sanderson M. Smith
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There are ten slots, and the probability of filling each slot is 1/6. We have N = 10 and p = 1/6.
The main purpose of this paper is to illustrate that the simple binomial formulas for mean, mean, and standard deviation will yield the same results that we would get by using the corresponding random variable formulas for the random variable x, where x is an element of the set {0,1,2,3,4,5,6,7,8,9,10}.
Here are the binomial calculations:
Now the computations will be done using random variable formulas
The computations in the following table were done on the TI83. If you put the values of x in list L1, then the probabilities are easily calculated in a second list using (10 nCr L1)*(1/6)^L1*(5/6)^(10L1) or binompdf(10,1/6,L1)





0.16151 
0.00000 
0.44863 

0.32301 
0.32301 
0.14356 

0.29071 
0.58142 
0.03230 

0.15505 
0.46514 
0.27564 

0.05427 
0.21706 
0.29545 

0.01302 
0.06512 
0.14371 

0.00217 
0.01302 
0.04076 

0.00025 
0.00174 
0.00706 

0.00002 
0.00015 
0.00075 

0.00000 
0.00001 
0.00004 

0.00000 
0.00000 
0.00000 
TOTALS 
1.00000 
1.66667 
1.38889 

This column total is 100%, as should be expected. 
This column total is the mean of the random variable x. Note that this is the same number obtained using the binomial mean formula. 
This column total is the variance of the random variable x. Note that this is the same number obtained using the binomial variance formula. 
A binomial setting is extremely nice since the formulas for calculating mean, standard deviation, and variance are simple, and easy to use. The table above demonstrates that one obtains the same values from random variable formulas that are necessary to use when a binomial setting does not exist.
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