In a large reserve forest spanning over $500$ square km$,$ the forest departments have adopted the

following scheme for counting the number of tigers $($N$).$

In the first week, an extensive search is conducted till the first tiger is sighted. Once the tiger is sighted,

the forest department has mechanism to catch the tiger without causing any harm to it$.$ Subsequently,

tranquillizer is used to make it fall asleep for a while, an identifying tag is permanently attached to the

tiger before releasing it to back to the forest.

In the second week, again an extensive search is conducted till a tiger is sighted. If this is found to be

the one caught earlier $($detected from the tag$),$ the search process stops immediately. Otherwise, the

same process of tagging is done with this tiger, as was the case in the first week.

The search continues in the third week $($if it was not decided to be stopped at the second week$).$ If this

is found to be one of the tigers caught earlier $($detected from the tag$),$ the search process stops

immediately. Otherwise, the same process of tagging is done with this tiger, as was the case with the

tiger caught in the first or the second week.

The search process continues following this process till it stops as per the stated guideline already

communicated. Assume that the search team is successful in sighting a tiger in every week it decided

to conduct the search. And only one tiger is tagged in every week of search.

Let X denote the number of weeks for which the search is conducted. It is believed that it is possible

to estimate N from observed value of X$.$ For this problem, however, we have a much simpler problem

assigned to you. Suppose N $=5.$

a$)$ Obtain the probability distribution of X$.$

b$)$ Find also the expected number of weeks during which the search is conducted. What is the

standard deviation of X$?$

c$)$ What is the chance of X being within two standard deviation of mean in this given case? Show

how it violates or confirms Chebyshev$$s theorem.