The Joker has a 20% chancer of killing a henchman during a crime. (They are easi

Question

The Joker has a 20% chancer of killing a henchman during a crime. (They are easily replaced, so subsequent crimes are independent. He never kills more than one henchman during a crime; he' s just trying to make a point.) Every time he commits a crime, he's caught, sent to Arkham Asylum, and escapes to commit another crime. After he has killed 8 henchmen, he will receive the death penalty.

a) Find the probability that the Joker lets (all) his henchmen live on exactly 25 crimes before he is executed..

b) Let X = the number of crimes where (all) the Joker's henchmen live . Write down the moment generating function , expected value, variance, and standard deviation of X.

Explanation / Answer

1. p=0.20 q=0.80

since it is said that, after he has killed 8 henchmen, he will receive the death penalty.so he couldnot kill 17 henchmen and killded 8 henchmen on his 18th to 25th crimes,

probability= 25C8 .(p)8 .(q)17

25C8 .(0.20)8 .(0.8)17

The moment generating function of X

is M(t) = E(etX) = (1-p)+pet

E(X)=p   V(X)= p(1-p)

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