This is a problem on the joint distribution of a discrete and a continuous random variable.
Kitty Oil Co. has decided to drill for oil in 10 different locations;
the cost of drilling at each location is 10,000. (Total cost is
then 100,000.)The probability of finding oil in a given location is only 0.2, but
if oil is found at a given location, then the amount of money the
company will get selling oil (excluding the initial 10,000 cost) from
that location is an exponential random variable with mean 50,000.Let Y be the random variable that denotes the number of locations
where oil is found, and let Z denote the total amount of money
received from selling oil from the locations.
Here what is expectation of $Z$ and $\mathbb{P}(Z>10000|Y=1)$?
How should one evaluate such questions in general??
Thanks in advance
Best Answer
It is not clear whether $Z$ is income or net income. No big deal, if we can handle one we can handle the other. We use the gross income interpretation.
Let $Z_1,Z_2,\dots, Z_{10}$ be the amount of money made from digs $1,2,\dots,10$. Then $Z=Z_1+Z_2+\cdots+Z_{10}$. By the linearity of expectation, we have $E(Z)=E(Z_1)+\cdots +E(Z_{10})=10E(Z_1)$.
To find $E(Z_1)$, note that $Z_1=0$ with probability $1-p$, where $p$ is the probability of finding oil if one digs, currently unreadable. And given that the well was successful, the expectation is $50000$. Thus $E(Z_1)=(1-p)(0)+(p)(50000)$.
For the probability that $Z\gt 10000$ given $Y=1$, we just want the probability that an exponential with mean $50000$ is greater than $10000$.
Remark: If we interpret $Z$ as net income, for the expectation question subtract $100000$.
For the probability question, find the probability that an exponential with mean $50000$ is greater than $110000$.