Answer:
a) For this case the random variable X follows a hypergometric distribution.
b) ![E(X)= n\frac{M}{N}=10 \frac{5}{25}=2](https://tex.z-dn.net/?f=E%28X%29%3D%20n%5Cfrac%7BM%7D%7BN%7D%3D10%20%5Cfrac%7B5%7D%7B25%7D%3D2)
![Var(X)=n \frac{M}{N}\frac{N-M}{N}\frac{N-n}{N-1}=10\frac{5}{25}\frac{25-5}{25}\frac{25-10}{25-1}=1](https://tex.z-dn.net/?f=Var%28X%29%3Dn%20%5Cfrac%7BM%7D%7BN%7D%5Cfrac%7BN-M%7D%7BN%7D%5Cfrac%7BN-n%7D%7BN-1%7D%3D10%5Cfrac%7B5%7D%7B25%7D%5Cfrac%7B25-5%7D%7B25%7D%5Cfrac%7B25-10%7D%7B25-1%7D%3D1)
c)
d)
Step-by-step explanation:
The hypergometric distribution is a discrete probability distribution that its useful when we have more than two distinguishable groups in a sample and the probability mass function is given by:
Where N is the population size, M is the number of success states in the population, n is the number of draws, k is the number of observed successes
The expected value and variance for this distribution are given by:
![E(X)= n\frac{M}{N}](https://tex.z-dn.net/?f=E%28X%29%3D%20n%5Cfrac%7BM%7D%7BN%7D)
![Var(X)=n \frac{M}{N}\frac{N-M}{N}\frac{N-n}{N-1}](https://tex.z-dn.net/?f=Var%28X%29%3Dn%20%5Cfrac%7BM%7D%7BN%7D%5Cfrac%7BN-M%7D%7BN%7D%5Cfrac%7BN-n%7D%7BN-1%7D)
a. What is the distribution of X?
For this case the random variable X follows a hypergometric distribution.
b. Compute the values for E(X) and Var(X)
For this case n=10, M=5, N=25, so then we can replace into the formulas like this:
![E(X)= n\frac{M}{N}=10 \frac{5}{25}=2](https://tex.z-dn.net/?f=E%28X%29%3D%20n%5Cfrac%7BM%7D%7BN%7D%3D10%20%5Cfrac%7B5%7D%7B25%7D%3D2)
![Var(X)=n \frac{M}{N}\frac{N-M}{N}\frac{N-n}{N-1}=10\frac{5}{25}\frac{25-5}{25}\frac{25-10}{25-1}=1](https://tex.z-dn.net/?f=Var%28X%29%3Dn%20%5Cfrac%7BM%7D%7BN%7D%5Cfrac%7BN-M%7D%7BN%7D%5Cfrac%7BN-n%7D%7BN-1%7D%3D10%5Cfrac%7B5%7D%7B25%7D%5Cfrac%7B25-5%7D%7B25%7D%5Cfrac%7B25-10%7D%7B25-1%7D%3D1)
c. What is the probability that none of the animals in the second sample are tagged?
So for this case we want this probability:
d. What is the probability that all of the animals in the second sample are tagged?
So for this case we want this probability: