The sum of the mean and variance of a binomial distribution is 15 and the sum of their squares is 117. The mean of the distribution is

The sum of the mean and variance of a binomial distribution is 15 and the sum of their squares is 117. The mean of the distribution is

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    Solution:

    Let n and p be parameters of the binomial distribution. Then,

    np+npq=15 and (np)2+(npq)2=117     [Given]

     n2p21+q2(np+npq)2=117152

     1+q2(1+q)2=1172251+q2(1+q)2=1325 251+q2=131+q2+2q12q2-26q+12=0 6q213q+6=02q-33q-2=0q=23 and p=1-q=1-23

     and p=13

     np+npq=15n×13+n×29=15 5n9=15n=27

    Hence, mean = np = 9.

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