Answer :
Answer:
Explained below.
Step-by-step explanation:
The random variable X is defined as the number of stars in a given volume of space.
[tex]X\sim \text{Poisson}\ (\lambda=1)[/tex]
The probability mass function of X is:
[tex]p_{X}(x)=\frac{e^{-\lambda}\lambda^{x}}{x!}[/tex]
(7)
Compute the probability of exactly two stars in 16 cubic light-years as follows:
[tex]P(X=2)=\frac{e^{-1}\times 1^{2}}{2!}=\frac{e^{-1}}{2}=\frac{0.36788}{2}=0.18394\approx 0.184[/tex]
(8)
Compute the probability of three or more stars in 16 cubic light-years as follows:
[tex]P(X\geq 3)=1-P(X<3)\\\\=1-P(X=0)-P(X=1)-P(X=2)\\\\=1-\sum\limits^{2}_{x=0}[\frac{e^{-1}\times 1^{x}}{x!}]\\\\=1-0.36788-0.36788-0.18394\\\\=0.0803[/tex]
(9)
In 16 cubic light years there is only 1 star.
Then in 1 cubic light years there will be, (1/16) stars.
Then in 4 cubic light years there will be, 4 × (1/16) = (1/4) stars.
(10)
In 16 cubic light years there is only 1 star.
Then in 1 cubic light years there will be, (1/16) stars.
Then in t cubic light years there will be, [t × (1/16)] stars.