Hi, the answer is "You pay $11 for a movie ticket and some amount at the concession stand for a total of $28". We know this because the key words "total" before $28 dollars - 'total' stands for an equal sign in equations.
The unknown, or x, is the "some amount at the concession stand". This is an unknown value, but we do know it is added to $11 dollars for the movie ticket (which is a known value.)
Therefore, we have 11 (the movie ticket) plus (key word: and) some amount (meaning x) = (key word: total) 28
I'm assuming is the shape parameter and is the scale parameter. Then the PDF is
a. The expectation is
To compute this integral, recall the definition of the Gamma function,
For this particular integral, first integrate by parts, taking
Substitute , so that :
The variance is
The second moment is
Integrate by parts, taking
Substitute again to get
Then the variance is
b. The probability that is
which can be handled with the same substitution used in part (a). We get
c. Same procedure as in (b). We have
and
Then
The volume of a pyramid is one-third the volume of a prism.
The formula for the volume of a pyramid is V = 1/3Bh