12. Let \(\mathrm{X}\) be a continuous random variable with density function,

\(f\left( x \right)=\left\{ \begin{aligned}

& \,\,\frac{1}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0<x\le 1 \\

& \,\,\frac{5-x}{12}\,\,\,\,\,\,\,\,\,\,\,\,\,1<x<5 \\

& \\

& \,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \\

\end{aligned} \right.\)

1. Find \(\mathrm{P}(1 / 2<\mathrm{X}<3)\).
2. Find \(\mathrm{P}(2<\mathrm{X}<6)\).
3. Find the mean of \(X\), i.e. \(E(X)\).
4. Find the first quartile, \(Q_{1}\).
5. Find the third quartile, \(\mathrm{Q}_{3}\).

13. a. Toss a fair coin twice. Let A be the event "head on the first toss". Let B be the event "both tosses have the same outcome". Are A and B independent events?

Which independence (dependence) criterion will you use? Write all events and the probabilities associated with these events.

Show your calculations to prove independence or dependence.

Problem 14: Al, Bob and Clifford go out for coffee. They play the following game to determine which one of the three pays for all three coffees.

Round 1: First Al, then Bob, and finally Clifford flip a fair coin. If there is an "odd man" i.e. odd man with a tail while the other two show a head, or odd man with a head while the other two show tail, the odd man pays the check

Round 2: If all three show a head or if all three show a tail, repeat the sequence as in round 1.

Additional rounds are played until there is an odd man. Let X be the number of rounds until the odd man is determined. What kind of special random variable is X ?

Price: $8.7

Solution: The downloadable solution consists of 5 pages, 370 words.
Deliverable: Word Document