Question

Let X be a continuous random variable such that 𝑓(𝑥) =1, 𝑓or 𝑓 0<𝑋<1.

Find the following probabilities:

1. 𝑃(𝑋>0.5)
2. 𝑃(0.5<𝑋<0.8)

Answer 1

1.

$$P(X>0.5)=1-P(X\le0.5)$$

$$=1-\displaystyle\int_{-\infin}^{0.5}f(x)dx=1-\displaystyle\int_{0}^{0.5}1dx$$

$$=1-[x]\begin{matrix} 0.5\\ 0 \end{matrix}=0.5$$

2.

$$P(0.5<X<0.8)=\displaystyle\int_{0.5}^{0.8}f(x)dx$$

$$=\displaystyle\int_{0.5}^{0.8}1dx=[x]\begin{matrix} 0.8\\ 0.5 \end{matrix}=0.3$$