Question

10.             A continuous random variable X having values only between 0 and 4 has density function given by f (x) =  1/2 - ax ; where “a”

is constant. Calculate “a”. Find P(1 < X < 2) 

Answer 1

i.

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{4}(\dfrac{1}{2}-ax)dx$$

$$=[\dfrac{x}{2}-\dfrac{ax^2}{2}]\begin{matrix} 4 \\ 0 \end{matrix}=2-8a=1$$

$$a=\dfrac{1}{8}$$

ii.

$$P(1<X<2)=\displaystyle\int_{1}^{2}(\dfrac{1}{2}-\dfrac{1}{8}x)dx$$

$$=[\dfrac{x}{2}-\dfrac{x^2}{16}]\begin{matrix} 2 \\ 1 \end{matrix}=1-\dfrac{1}{4}-(\dfrac{1}{2}-\dfrac{1}{16})=\dfrac{5}{16}$$