Question

6) Diagram of a probability density function is shown below. f(t) 7 5 10 11 12 Time (hours) a. Derive the mathematical expression for the density function f(t). (6 marks) b. Find P (3<t<7). (3 marks)

Answer 1

a) The height of the triangle is first computed by using the property that the area under a PDF should be 1. Therefore, we have here:

0.5*12*H = 1

H = 1/6 here.

The first line of PDF goes from (0,0) to (4, 1/6)
The equation of line here is obtained as:
f(t) - 0 = (t - 0)(1/24)
f(t) = t/24 for the first part here.

For second part, the points given as: (4, 1/6) and (12, 0)
Therefore, the equation here is obtained as:
f(t) - 0 = (t - 12) (1/6)*(1/-8)

f(t) = (t - 12) (-1/48)

f(t) = (12 - t)/48

therefore the PDF for t here is obtained as:

$f(t) = \left\{\begin{matrix} t/24 & 0 < t < 4 \\ (12 - t)/ 48 & 4 < t < 12 \end{matrix}\right.$

b) The probability here is computed as:

$P(3 < t < 7) = \int_{3}^{4}\frac{t}{24} \ dt + \int_{4}^{7} \frac{(12 - t)}{48} \ dt$

$P(3 < t < 7) = \frac{4^2 - 3^2}{48} + \frac{(12*3 - 0.5(7^2 - 4^2))}{48} = 0.5521$

Therefore 0.5521 is the required probability here.