Answer:
Step-by-step explanation:
The model  , the number of planets found up to time
, the number of planets found up to time  , as a Poisson process. So, the
, as a Poisson process. So, the  has distribution of Poison distribution with parameter
 has distribution of Poison distribution with parameter 
a)
The mean for a month is,  per month
 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
. The  sum  of  of these  independent exponential has distribution Gamma
 of these  independent exponential has distribution Gamma  .
.
From the given information,  and
 and 
Calculate the expected value.

(Here,  and
 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)

Hence, the required probability is 0.2149 or 21.49%