Question

A continuous random variable has a density function𝑓(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 ≤ 𝑥 ≤ 2.7)

Answer 1

I) p(x<2.5)= $\displaystyle\int\limits^{{2.5}}_{{2}} 2\cdot\dfrac{(5-x)}{5}\,{{d}x}$

= $\int_{2}^{2.5}2dx-\int_{2}^{2.5}\frac{2x}{5}dx$

$=2x-\frac{x^2}{5}| 2 to 2.5$

$=\frac{11}{20}$

ii)$\displaystyle\int\limits^{{3}}_{{2.2}} 2\cdot\dfrac{(5-x)}{5}\,{{d}x}$

$\int_{2.2}^{3}2dx-\int_{2.2}^{3}\frac{2x}{5}dx$

$=\frac{96}{125}$

iii)$\displaystyle\int\limits^{{2.7}}_{{2.1}} 2\cdot\dfrac{(5-x)}{5}\,{{d}x}$

$\int_{2.1}^{2.7}2dx-\int_{2.1}^{2.7}\frac{2x}{5}dx$

$=\dfrac{78}{125}$