Question

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

a. Compute the variance.

Answer 1

Answer on Question #51084 – Math – Statistics and Probability

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

a. Compute the variance.

Solution

First case (continuous uniform distribution)

Assume that the time is continuous random variable

If the time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes, then the probability density function is given by

$$f(x) = \begin{cases} 0 & \text{for } x < a \\ \frac{1}{b - a} & \text{for } a \leq x \leq b \\ 0 & \text{for } x > b \end{cases} = \begin{cases} 0 & \text{for } x < 110 \\ \frac{1}{190 - 110} & \text{for } 110 \leq x \leq 190 \\ 0 & \text{for } x > 190 \end{cases} = \begin{cases} \frac{1}{80} & \text{for } 110 \leq x \leq 190 \\ 0 & \text{for } x > 190 \end{cases}$$

the mean is given by $m = (110 + 190)/2 = 150$

the variance is given by

$$Var(X) = \frac{(b - a)^2}{12} = \frac{(190 - 110)^2}{12} = \frac{80^2}{12} = \frac{6400}{12} = 533.333$$

Second case (discrete uniform distribution)

Assume that the time is discrete random variable.

X = x 110 190

P(X = x) 0.5 0.5

If the time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes, then first we need to find

the mean $m = (110 + 190)/2 = 150$

and finally compute

the variance $Var(X) = 0.5(110 - 150)^2 + 0.5(190 - 150)^2 = 1600$

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