2.II.11F

Dipkomsky, a desperado in the wild West, is surrounded by an enemy gang and fighting tooth and nail for his survival. He has $m$ guns, $m>1$, 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 $1 / 2$ and moves to the next one. Otherwise, i.e. when the gun is unloaded, he loads it with probability $3 / 4$ or simply moves to the next gun with complementary probability $1 / 4$. If he decides to load the gun he then fires it or not with probability $1 / 2$ and after that moves to the next gun anyway.

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

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