Question

a random variable x observes the following density in the interval [0, 1] f(x) = { a + bx, for 0 ≤ x ≤ 1 0, elsewhere .for what value of a and b does the median associated with the above density equal 1/2

Answer 1

$$f(x) = \begin{cases} a+bx &0\leq x\leq 1 \\ 0 &elsewhere \end{cases}$$

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{1}(a+bx)dx=[ax+\dfrac{1}{2}bx^2]\begin{matrix} 1 \\ 0 \end{matrix}$$

$$=a+\dfrac{1}{2}b=1$$

$$\displaystyle\int_{0}^{1/2}(a+bx)dx=[ax+\dfrac{1}{2}bx^2]\begin{matrix} 1/2 \\ 0 \end{matrix}$$

$$=\dfrac{1}{2}a+\dfrac{1}{8}b=\dfrac{1}{2}$$

$$a+\dfrac{1}{2}b=1$$

$$a+\dfrac{1}{4}b=1$$

$$a=1, b=0$$