Question

A random variable X, the time taken by a garage to service a car, takes values between 0 and 10

hours with cumulative distribution function. F(x)= A + B In (3x + 2) for 0≤ x ≤ 10.

Find the values of A and B.

Answer 1

$F(x)= A + Bln(3x + 2) \\ F(0) = A + Bln(2) = 0 \\ F(10) = A +Bln(32) = 1 \\ A = -Bln(2) \\ -Bln(2) +Bln(32) = 1 \\ B(ln(32) -ln(2)) = 1 \\ B = \frac{1}{ln(32) -ln(2)} = \frac{1}{3.465 -0.693}= \frac{1}{2.772} = 0.361 \\ A = -0.361 \times 0.693 = -0.25$