Question

Suppose the pdf of the magnitude of X of a dynamic load on a bridge is given by ��={ 1 8(1+3�,0≤�ɤ2 0, �ѡℎ�џ�і�ђ i) Determine �0.5≤�ɤ1.5) ii) Obtain the cumulative density function of X

Answer 1

$$f(x)=\begin{matrix} 1/8+(3/8)x & 0\leq x\leq 2 \\ \\ 0 & otherwise \end{matrix}$$

i)

$$P(0.5\leq x\leq 1.5)=\displaystyle\int_{0.5}^{1.5}(1/8+(3/8)x)dx$$

$$=\bigg[(1/8)x+(3/16)x^2\bigg]\begin{matrix} 1.5 \\ 0.5 \end{matrix}$$

$$=\dfrac{1.5}{8}+\dfrac{3(1.5)^2}{16}-\dfrac{0.5}{8}-\dfrac{3(0.5)^2}{16}=0.5$$

ii)

$$F(x)=\displaystyle\int_{-\infin}^{x}f(x)dx$$

$$F(x) = \begin{cases} 0 &x<0 \\ (1/8)x+(3/16)x^2 &0\leq x<2 \\ 1 & x\geq 2 \end{cases}$$