Probability | Part IA, 2001

Dipkomsky, a desperado in the wild West, is surrounded by an enemy gang and fighting tooth and nail for his survival. He has mm guns, m>1m>1, pointing in different directions and tries to use them in succession to give an impression that there are several defenders. When he turns to a subsequent gun and discovers that the gun is loaded he fires it with probability 1/21 / 2 and moves to the next one. Otherwise, i.e. when the gun is unloaded, he loads it with probability 3/43 / 4 or simply moves to the next gun with complementary probability 1/41 / 4. If he decides to load the gun he then fires it or not with probability 1/21 / 2 and after that moves to the next gun anyway.

Initially, each gun had been loaded independently with probability pp. Show that if after each move this distribution is preserved, then p=3/7p=3 / 7. Calculate the expected value ENE N and variance Var NN of the number NN of loaded guns under this distribution.

[Hint: it may be helpful to represent NN as a sum 1jmXj\sum_{1 \leq j \leq m} X_{j} of random variables taking values 0 and 1.]

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