Question

The time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes.

a. what is the probability that a guitar neck can be carved between 95 and 165 minutes?

b. what is the probability that the guitar neck can be carved between 120 and 200 minutes?

c. determine the expected completion time for carving the guitar neck.

d. compute the standard deviation.

Answer 1

The probability density function of a uniform distribution is given by :

1/(b - a)

In our case the pdf is :

1/(190 - 110)

= 1/80

a) ₉₅∫¹⁶⁵(1/80) dt

1/80[t] |t = 95 - 165

165/80 - 95/80 = 7/8

= 7/8

b) ₁₂₀∫²⁰⁰(1/80)dt

1/80[t] | t = 120 - 200

200/80 - 120/80 = 1

= 1

c)E(t)

= ₁₁₀∫¹⁹⁰t1/80 dt

1/80[t²/2] | t = 110 to 190

= 18050/80 - 6050/80 = 150

= 150 minutes

d) Variance

E(t²)

= ₁₁₀∫¹⁹⁰t²1/80 dt

1/80[t³/3] | t = 110 to 190

= 28579.17 - 5545.83

= 23033.34

Standard deviation = √23033.34 = 151.77