 Hi and welcome to the session. Let's work out the following question. The question says that the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution. So let's start with the solution to this question. Let N and P be the parameters of binomial distribution, then we will have mean plus variance is equal to 9 plus 6, that is given to us in the question. Now this implies that N P plus N P Q is equal to 15 and this implies that N P into 1 plus Q is equal to 15 and this implies N P is equal to 15 divided by 1 plus Q. We call this equation 1 and we see that mean into variance will be equal to 9 into 6. This implies N P into N P Q is equal to 54 and this implies that N square P square Q is equal to 54. This implies N square P square is equal to 54 divided by Q and we call this equation 2. Now from 1 and 2 we get 15 divided by 1 plus Q the whole square is equal to 54 divided by Q. This implies 225 divided by 1 plus Q square plus 2 Q is equal to 54 divided by Q. This implies 225 Q is equal to 54 into 1 plus 2 Q plus Q square. This we get by cross multiplication. This implies 54 Q square minus 117 Q plus 54 is equal to 0. This implies that Q is equal to 117 plus minus square root of 117 the whole square minus 4 point into 51 into 51 because this is A, this is B and this is C and we see that this will be equal to minus B plus minus square root B square minus 4 AC divided by 2 into A that is 2 multiplied by 54. This implies that Q is equal to 117 plus minus square root of 2025 divided by 108. This implies that Q is equal to 117 plus minus 45 divided by 108. This implies that Q is equal to 162 by 108 or 72 by 108. This implies Q is equal to 3 by 2 or 2 by 3 but we see that Q is equal to 3 by 2 is greater than 1 which is not possible because P plus Q should be equal to 1 therefore Q is equal to 2 by 3. Also P will be equal to 1 minus Q that is 1 minus 2 by 3 and that is equal to 1 by 3. Now we see that mean is equal to N into P. Now mean is given to be 9. P is 1 by 3 so we have 1 by 3 into N. This implies that N is equal to 9 into 3 that is 27. So our answer to this question is N is equal to 27, E is equal to 1 by 3 and Q is equal to 2 by 3. So this is our answer to this question. I hope that you understood the solution and enjoyed the session. Have a good day.