Chapter 3.1, Problem 3.7E

(a)

To determine

To explain the reason due to which the height of the curve is equal to 1.

Explanation of Solution

Given:

  

The height of the given graph is 1, because it is equally distributed for the range 0 − 1 as 1 and the area under the curve is equal to 1. The height of the uniform distribution defines the proportion of the area per 1 unit and since here the interval is of 1 unit only so the height is also 1.

(b)

To determine

To calculate the mean of the density curve.

Answer to Problem 3.7E

Mean = 0.5

Explanation of Solution

Given:

  $x~\text{Uniform(0, 1)}$

Formula used:

  $\text{Mean = }\frac{\text{1}}{2}\left(a+b\right)$

Calculation:

  $\begin{array}{c}\text{Mean = }\frac{\text{1}}{2}\left(a+b\right)\\ =\frac{1}{2}\left(0+1\right)\\ =0.5\end{array}$

(c)

To determine

To calculate the mean of the density curve.

Answer to Problem 3.7E

Median = 0.5

Explanation of Solution

Calculation:

  $\begin{array}{c}\text{Median }\Rightarrow {\displaystyle {\int }_0^x\frac{1}{1-0}dx}=0.5\\ \Rightarrow 1{\left[x\right]}_0^x=0.5\\ \Rightarrow \left[x-0\right]=0.5\\ \Rightarrow x=0.5\end{array}$

(d)

To determine

To calculate the proportion that falls between 0 and 0.4.

Answer to Problem 3.7E

0.4

Explanation of Solution

Calculation:

  $\begin{array}{c}\text{Proportion }\Rightarrow {\displaystyle {\int }_0^{0.4}\frac{1}{1-0}dx}\\ \Rightarrow 1{\left[x\right]}_0^{0.4}\\ \Rightarrow \left[0.4-0\right]\\ \Rightarrow 0.4\end{array}$