Answer:
a) 0.264 b) 64 cubic light years
Step-by-step explanation:
Let be X the following event : ''Number of stars in a given volume of space''
X~Poisson variable
X~P(k;λ)
The density is 1 star per 16 cubic light years so
λ = 1 star / 16 cubic light years
The probability mass function of X is :
f(x)= P(X = k) = {[(λt) ^ k].[(e) ^ (-λt)]} / k! ; k∈Νo
We measure t in units of 16 cubic light years
λt = (1 star / 16 cubic light years).(1).(16 cubic light years) = 1 star
a)
[tex]P(X\geq 2)= 1-P(X<2)=1-P(X=0)-P(X=1)[/tex]
[tex]P(X\geq 2)=1-f(0)-f(1)[/tex]
[tex]P(X\geq 2) = 1-e^{-1}-e^{-1} =0.264[/tex]
b) We are looking for t so that :
[tex]P(X\geq1)\geq 0.98\\1-P(X<1)\geq 0.98\\1-f(0)\geq 0.98\\1-e^{-t} \geq 0.98\\-e^{-t} \geq -0.02[/tex]
[tex]e^{-t} \leq 0.02\\ln(e^{-t}) \leq ln(0.02)\\-t\leq ln(0.02)\\t\geq -ln(0.02)\\t\geq 3.912[/tex]
t≥3.912 ≈4
We should explore 4.16= 64 cubic light years of space to find a star with the certain of 98%.
