Question

The mean of the binomial distribution 10∁x ( 2/3 )^x ( 1/3 ) ^ 10 -x is given by

Answer 1

The probability mass function of binomial dsitribution is given by

=

Where x can take only integer values from 0 to 10.

Now, this probability mass function actually implies that X follows binomial distribution with parameters n=10 and p=2/3.

Now, we know that the mean of a binomial distribution with parameters n and p is given by

.

Now, we use that proof, to find the mean of X.

ie.

ie.

ie.

ie.

Now, by the binomial theorem, we know that the binomial expansion is given by

So, according to this rule, the part under summation in the last written part of the equation becomes

ie. 1.

So, the mean, ie. E(X) becomes

ie.

So, the mean of this binomial distribution is 20/3.

Note:- I have given the proof but you can always skip it, using the fact that the mean of a binomial distribution with parameters n and p, is n*p, and the variance is n*p*q. Here also, you could identify n=10 and p=2/3, and just multiply them, to get the mean=20/3.