![5. Assume initially, 100 molecules are inside a cell. After a long time, we found molecule is inside the cell with probability 1/4. Let the random variable M he number of molecules in the cell after long time. (a) [Extra 3 points] What are the conditions for M to have a binomial distribution? M (b) [5 points] Under the conditions in (a) find the mean and mode of](http://img.homeworklib.com/questions/a1df0720-9bfe-11ec-ba3d-dd833caa5158.png?x-oss-process=image/resize,w_560)
Homework Answers
1) M is binomial variable when it is descete and varies over the range 0 to n that is 0 to 100.
2) p=p(success)=1/4
Mean=n*p=100*1/4=2
Binomial distribution has 2 modes which are integer part of (n+1)*p and integer part of (n+1)*p-1.
Hence in this case mode is 25 and 24 respectively according to the above formulae.
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