84 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



pected number, E(r), for any designated future trial. Hence it al- 

 ready has been found from experience and intuition that for the 

 binomial situation ^ = np. This result can be established for any 

 binomial frequency distribution, but such will not be done herein. 



It also can be shown by somewhat more difficult mathematics that 

 the standard deviation for a binomial frequency distribution is given 

 by a = vnpq ; consequently, given n and p, we can compute the mean 

 and standard deviation very easily. This will be found to be helpful 

 later in this chapter. 



Sometimes when dealing with binomial distributions it is advanta- 

 geous to work with the fractional number of occurrences, r/n, rather 

 than with the actual number, r. In this case, the arithmetic mean is 

 p, rather than np; and the standard deviation is vpq/n , instead of 

 vnpq. To illustrate the use of these formulas both for r and for r/n 

 consider again the insurance example above in which n = 10 and p 

 = .1. Under these circumstances the mean r is np = 10(.l) = 1, and 

 the standard deviation is vnpq = vl0(.l)(.9) = 0.949, approxi- 

 mately. The mean r/n is p = .1, and the standard deviation of the 

 fraction dead is vpq/n = V (.1)(.9)/10 = 0.095, approximately. If 

 the number n were sufficiently large that it would be practical, such 

 information as that just derived might be useful to an insurance com- 

 pany in anticipating the average number (or fraction) of death benefits 

 it could expect to pay, and in making sufficient allowance for chance 

 deviations from those average numbers so that adequate funds would 

 be available to pay death benefits. 



PROBLEMS 



1. Use the coefficients of (l/2) r in the series for (1/2 + 1/2) 6 to graph the 

 binomial frequency distribution appropriate to sets of 6 trials with an event 

 whose probability of occurrence is constantly p = 1/2. 



2. Under the conditions of problem 1, what is the probability that the event 

 will occur at least 4 times on 6 trials. Ans. 11/32. 



3. Graph the frequency distributions for the binomial with p = 1/2 and 

 n = 4, 8, and 12, successively. Compute the /jl and a in each instance and locate 

 on the scale of r: /j. ± lcr, fi ± 2u, and fi ± 3o\ 



4. Graph the binomial frequency distribution for p = 1/4, n = 4, and read 

 from it the probability that r will be 2, 3, or 4. 



5. Check the result obtained in problem 4 by constructing the r.c.j. graph and 

 reading the answer from this graph. 



6. Graph the binomial frequency distribution for p = .7, n = 4, and determine 

 the probability that on one random set of 4 trials E will occur at least fi times, 

 where n = arithmetic mean. 



