Question

Suppose that the random variable X is the time taken by a garage to service a car: These times are distributed between 0 and 10 hours with a cumulative distribution function given

F(x) = -0.25+ 0.361 ln(3x+ 2)-0.25 for 0<=x<=10. Find the probability density function

Answer 1

Explanation:

To find the probability density function (PDF) from the given cumulative distribution function (CDF), we need to take the derivative of the CDF with respect to x.

Given CDF:

F(x) = -0.25 + 0.361 ln(3x + 2) - 0.25 for 0 <= x <= 10

Step 1: Take the derivative of the CDF with respect to x:

f(x) = d/dx [F(x)]

Step 2: Find the PDF f(x) for 0 <= x <= 10.

For 0 <= x <= 10, the CDF F(x) is given by:

F(x) = -0.25 + 0.361 ln(3x + 2) - 0.25

Now, take the derivative of F(x) with respect to x to find the PDF f(x):

f(x) = d/dx [-0.25 + 0.361 ln(3x + 2) - 0.25]

Step 3: Simplify the derivative:

f(x) = 0 + 0.361 * d/dx [ln(3x + 2)] - 0

The derivative of ln(3x + 2) with respect to x is:

d/dx [ln(3x + 2)] = 1 / (3x + 2) * d/dx [3x + 2] = 1 / (3x + 2) * 3

f(x) = 0.361 * 3 / (3x + 2) = 1.083 / (3x + 2)

So, the probability density function (PDF) of the random variable X is given by:

f(x) = 1.083 / (3x + 2) for 0 <= x <= 10.