Jack and Jill independently arrange the digits 1 to 9 in any order and compare the resulting numbers digit by digit. Let X denote the number of digits in agreement.
For example, if the numbers were 132456789 and 142567893, X = 2 as the first and third digits are the same.
Find E[X] and Var[X]
My work:
I'm having difficulty figuring out the distribution of X. I thought X might be negative binomial distributed with X ~ NBin (n, 1/9). Only problem with this is that the 1/9 changes depending on whether the previous digit was a success or failure. I think it might be negative binomial involving some summations but I don't know what
Then I tried by brute force, I know there are 9! different ways to arrange the numbers 1 to 9 but I really don't want to calculate the P(X) = 0,1,2 and so forth. Is there an easier way to do this question?
And even if I find the expectation, I have no idea how to find E[X]^2 to find the variance.
For example, if the numbers were 132456789 and 142567893, X = 2 as the first and third digits are the same.
Find E[X] and Var[X]
My work:
I'm having difficulty figuring out the distribution of X. I thought X might be negative binomial distributed with X ~ NBin (n, 1/9). Only problem with this is that the 1/9 changes depending on whether the previous digit was a success or failure. I think it might be negative binomial involving some summations but I don't know what
Then I tried by brute force, I know there are 9! different ways to arrange the numbers 1 to 9 but I really don't want to calculate the P(X) = 0,1,2 and so forth. Is there an easier way to do this question?
And even if I find the expectation, I have no idea how to find E[X]^2 to find the variance.