Question

For the function $f\left(x\right)=a+bx,0\le x\le 1,$ to be a valid probability density function, which one of the following statements is correct?

• A. a = 1, b = 4
• B. a = 0.5, b = 1
• C. a = 0, b = 1
• D. a = 1, b = -1

Answer 1

The correct option is B a = 0.5, b = 1

For probability density function f(x) to be valid

${\int }_{-\infty }^{\infty }f\left(x\right)dx=1$

${\int }_{-\infty }^{\infty }(a+bx)dx=1$

${\int }_0^1(a+bx)dx=1$

${[ax+\frac{b{x}^2}{2}]}_0^1=1$

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

for equation to be satisfied

a = 0.5 b = 1