Question

A. the time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes. 1. what is the probability that a guitar neck can be carved between 95 and 165 minutes? 2. what is the probability that the guitar neck can be carved between 120 and 200 minutes? 3. determine the expected completion time for carving the guitar neck. 4. compute the variance.

Answer 1

The pdf of this law is given by the formula :
$\dfrac{1}{190-110}\text{ if } 110\leq x\leq190 \text{ and 0 otherwise.}$
We will use the above formula in order to compute the probability like this:
$\dfrac{1}{80}\int_{95}^{165}dx=\dfrac{1}{80}\int_{110}^{165}dx\\ =\dfrac{1}{80}(165-110)\\=\dfrac{55}{80}$
Similarly, we find the second probability (question 2) like this:
$\frac{190-120}{80}= \frac{70}{80}=\frac{7}{8}$.
The variance is given by the formula:
$\frac{1}{12}80^2 =\frac{800}{12}$