Question

The life of a manufacturer's LED lightbulbs is estimated by the probability distribution function:

f(x) = { 0 for x < 0; 1/30 for 0 ≤ x < 30; 0 for x ≥ 30.
Find the cumulative distribution function (CDF) F(x) for this distribution.

Answer 1

The CDF for the given PDF can be found by integrating the PDF over the given range. The CDF function F(x) is given by F(x) = { 0 for x < 0; x/30 for 0 ≤ x < 30; 1 for x ≥ 30 }.

Step-by-step explanation:

The cumulative distribution function (CDF) for a probability distribution function (PDF) can be found by integrating the PDF over the given range. In this case, the PDF given is f(x) = { 0 for x < 0; 1/30 for 0 ≤ x < 30; 0 for x ≥ 30. To find the CDF, we need to integrate the PDF from negative infinity to x.

For x < 0, P(X ≤ x) = ∫[negative infinity to x] f(t) dt = 0

For 0 ≤ x < 30, P(X ≤ x) = ∫[negative infinity to x] f(t) dt = ∫[0 to x] (1/30) dt = x/30

For x ≥ 30, P(X ≤ x) = ∫[negative infinity to x] f(t) dt = ∫[0 to 30] (1/30) dt = 1

So, the cumulative distribution function (CDF) F(x) for this distribution is:

F(x) = { 0 for x < 0; x/30 for 0 ≤ x < 30; 1 for x ≥ 30 }