The red-billed tropicbird is a bird of intertropical regions.
1. When the red-billed tropicbird lives in a polluted environment, its lifespan, in years, is modelled by a random variable $X$ following a normal distribution with unknown mean $\mu$ and standard deviation $\sigma = 0.95$.
a. Consider the random variable $Y$ defined by $Y = \frac { X - \mu } { 0.95 }$.
Give without justification the distribution followed by the variable $Y$.
b. It is known that $P ( X \geqslant 4 ) = 0.146$.
Prove that the value of $\mu$ rounded to the nearest integer is 3.
2. When the red-billed tropicbird lives in a healthy environment, its lifespan, in years, is modelled by a random variable $Z$.
The curves of the density functions associated with the distributions of $X$ and $Z$ are represented in the APPENDIX to be returned with the answer sheet.
a. Which is the curve of the density function associated with $X$? Justify.
b. On the APPENDIX to be returned with the answer sheet, shade the region of the plane corresponding to $P ( Z \geqslant 4 )$.
It will be admitted henceforth that $P ( Z \geqslant 4 ) = 0.677$.
3. A statistical study of a given region established that $30\%$ of red-billed tropicbirds live in a polluted environment; the others live in a healthy environment.
A red-billed tropicbird living in the given region is chosen at random.
Consider the following events:
- $S$ : ``the red-billed tropicbird chosen lives in a healthy environment'';
- $V$ : ``the red-billed tropicbird chosen has a lifespan of at least 4 years''.
a. Complete the weighted tree illustrating the situation on the APPENDIX to be returned with the answer sheet.
b. Determine $P ( V )$. Round the result to the nearest thousandth.
c. Given that the red-billed tropicbird has a lifespan of at least 4 years, what is the probability that it lives in a healthy environment? Round the result to the nearest thousandth.