Question

A continuous random variable has a density function f(x)= 2(5-x)/5 , where 2<x<3. Calculate
the following probability correct up to 3 decimal places, and make the graph for part (a) only in Answer sheet:
P (x < 2.5)
P (x > 2.2)
P (2.1 ≤

Answer 1

$$P(x<2.5)=\displaystyle\int_{2}^{2.5}\dfrac{2(5-x)}{5}dx$$

$$=\dfrac{2}{5}\big[5x-\dfrac{x^2}{2}\big]\begin{matrix} 2.5 \\ 2 \end{matrix}=0.55$$

$$P(x>2.2)=1-P(x\leq2.2)=1-\displaystyle\int_{2}^{2.2}\dfrac{2(5-x)}{5}dx$$

$$=1-\dfrac{2}{5}\big[5x-\dfrac{x^2}{2}\big]\begin{matrix} 2.2 \\ 2 \end{matrix}=0.768$$

$$P(2.1\leq x\leq 2.7)=\displaystyle\int_{2.1}^{2.7}\dfrac{2(5-x)}{5}dx$$

$$=\dfrac{2}{5}\big[5x-\dfrac{x^2}{2}\big]\begin{matrix} 2.7 \\ 2.1 \end{matrix}=0.624$$