Please show step by step Every evening a local superhero goes to fight crime. Th

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Please show step by step

Every evening a local superhero goes to fight crime. The superhero will beat one villain at a time until he gets bored and goes home. Suppose that after extensive research you conclude that the probability that the superhero will get bored after defeating any single criminal is 10% independently of any others. i. Find the expected value and standard deviation of the number of criminals that the superhero will defeat in one night? ii, The Superhero Union has a requirement for its members that they have to defeat at least 3500 villains in a year. Estimate the probability that our superhero will stay a member this year.

Explanation / Answer

let X is number of criminals defeated by Superhero

from above P(X=1)=P(afterbeating one villain Hero gets bored) =0.1

P(X=2)=P(does not get bored after beating first but bored after second)=0.9*0.1=0.09

P(X=3)=0.9*0.9*0.1=0.081

....

,,,

therefore expected value E(X)=0.1*1+0.09*2+0.081*3+..... =0.1*(1+0.9*2+0.92*3+...) =0.1/(1-0.9)2 =10

E(X2)=0.1*12+0.09*22+0.081*32+..... =0.1*(1+0.9*4+0.92*9+...) =0.1*(1+0.9)/(1-0.9)3 =190

hence std deviation =sqrt(E(X2)-(E(X))2) =sqrt(190-102) =9.487

ii)

for a year expected villain defeated =365*10 =3650

and std deviation =9.487*sqrt(365)=181.25

hence from normal approximation and continuity correction:

probability that superhero will stay a member =P(X>=3500)=P(Z>(3499.5-3650)/181.25)

=P(Z>-0.83)=0.7967

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