Answer:
Step-by-step explanation:
The model
, the number of planets found up to time
, as a Poisson process. So, the
has distribution of Poison distribution with parameter ![(\lambda t)](https://tex.z-dn.net/?f=%28%5Clambda%20t%29)
a)
The mean for a month is,
per month
![E[N(t)]= \lambda t\\\\=\frac{1}{3}(24)\\\\=8](https://tex.z-dn.net/?f=E%5BN%28t%29%5D%3D%20%5Clambda%20t%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B3%7D%3C%2Fp%3E%3Cp%3E%2824%29%5C%5C%5C%5C%3D8)
(Here. t = 24)
For Poisson process mean and variance are same,
![Var[N (t)]= Var[N(24)]\\= E [N (24)]\\=8](https://tex.z-dn.net/?f=Var%5BN%20%28t%29%5D%3D%20Var%5BN%2824%29%5D%5C%5C%3D%20E%20%5BN%20%2824%29%5D%5C%5C%3D8)
(Poisson distribution mean and variance equal)
The standard deviation of the number of planets is,
![\sigma( 24 )] =\sqrt{Var[ N(24)]}=\sqrt{8}= 2.828](https://tex.z-dn.net/?f=%5Csigma%28%2024%20%29%5D%20%3D%5Csqrt%7BVar%5B%20N%2824%29%5D%7D%3D%5Csqrt%7B8%7D%3D%202.828%3C%2Fp%3E%3Cp%3E)
b)
For the Poisson process the intervals between events(finding a new planet) have independent exponential distribution with parameter
. The sum of
of these independent exponential has distribution Gamma
.
From the given information,
and ![\lambda =\frac{1}{3}](https://tex.z-dn.net/?f=%5Clambda%20%3D%5Cfrac%7B1%7D%7B3%7D%3C%2Fp%3E%3Cp%3E)
Calculate the expected value.
![E(x)=\frac{\alpha}{\beta}\\\\=\frac{K}{\lambda}\\\\=\frac{6}{\frac{1}{3}}\\\\=18](https://tex.z-dn.net/?f=E%28x%29%3D%5Cfrac%7B%5Calpha%7D%7B%5Cbeta%7D%5C%5C%5C%5C%3D%5Cfrac%7BK%7D%7B%5Clambda%7D%3C%2Fp%3E%3Cp%3E%5C%5C%5C%5C%3D%5Cfrac%7B6%7D%7B%5Cfrac%7B1%7D%7B3%7D%7D%5C%5C%5C%5C%3D18)
(Here,
and
)
C)
Calculate the probability that she will become eligible for the prize within one year.
Here, 1 year is equal to 12 months.
P(X ≤ 12) = (1/Г (k)λ^k)(x)^(k-1).(e)^(-x/λ)
![=\frac{1}{Г (6)(\frac{1}{3})^6}(12)^{6-1}e^{-36}\\\\=0.2148696\\=0.2419\\=21.49%](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B%D0%93%20%20%286%29%28%5Cfrac%7B1%7D%7B3%7D%29%5E6%7D%2812%29%5E%7B6-1%7De%5E%7B-36%7D%5C%5C%5C%5C%3D0.2148696%5C%5C%3D0.2419%5C%5C%3D21.49%25)
Hence, the required probability is 0.2149 or 21.49%