Answer to Question #126861 in Statistics and Probability for Jay

Question

A sawmill makes boards that are used in construction projects. The board lengths of two-by-fours produced by this sawmill are uniformly distributed from 6 ft to 12 ft.
Question 1: Describe the graph of this distribution.
Question 2: Determine the probability that a board selected at random will be at least 10 ft long.

Answer 1

Solution:

Q1: The probability density function and comutative distribution function on the interval [6;12] are P(x) and D(x) respectively.

$P(x) = \begin{cases} 0 &\text{if }x<6 \\ 1/6 &\text{if } 6\le x \le 12 \\ 0 &\text{if }x > 12 \end{cases}$

The graph P(x) is piecewise constant.

$D(x) = \begin{cases} 0 &\text{if }x<6 \\ (x-6)/6 &\text{if } 6\le x \le 12 \\ 1 &\text{if }x > 12 \end{cases}$

The graph D(x) is piecewise linear.

Q2: $P(X\ge 10) =D(+\infty)-D(10)=1-(10-6)/6=1/3$