SOLUTION: Two balls are dropped in such a way that each ball is equally likely to fall into any one of four holes. Both balls may fall into the same hole. Let X denote the number of unoccu

Question 1149656: Two balls are dropped in such a way that each ball is equally likely to
fall into any one of four holes. Both balls may fall into the same hole. Let X
denote the number of unoccupied holes at the end of the experiment. What
is the moment generating function of X?
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
2 balls can fall into 4 holes, therefore there are 4 * 4 = 16 ways that the balls can fall
:
We know that there can be either 3 holes unoccupied or 2 holes unoccupied
:
3 holes unoccupied means 2 balls in a hole, there are 4 ways to do this
:
Probability(P) (X = 3) = 4/16 = 1/4
:
P for 2 holes unoccupied can be calculated by subtracting P (X = 3) from 1, that is
:
P(X = 2) = 1 - P(X = 3) = 1 - 1/4 = 3/4
:
X is a discrete random variable, so the moment generating function for X is defined as
:
MX(t) = summation for X of e^Xt * f(x)
:
We can write the following for this problem
:
MX(t) = (3/4) * e^2t + (1/4) * e^3t
:
Note t is a parameter which does not relate to X, t is a dummy variable that could just as well be s or u
:

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