Jawaharlal Nehru Technological University Hyderabad B.Tech IV RELIABILITY ENGINEERING Previous Question Paper (July/August - 2022 - R16)

The Probability Density Function (PDF), denoted as f(t), represents the rate of change of the probability of failure with respect to time. It describes the likelihood of a component failing at a specific instant t. Mathematically, the area under the PDF curve over an interval gives the probability of failure within that time frame. For example, if a light bulb has a PDF of f(t) = 0.001e^(-0.001t), the value at t=1000 indicates the instantaneous failure density at that time.

The Probability Distribution Function (Cumulative Distribution Function - CDF), denoted as F(t), represents the cumulative probability of failure from time zero up to time t. It is defined as the integral of the PDF: F(t) = ∫ f(x) dx from 0 to t. F(t) gives the probability that the component will fail at or before time t. Using the same light bulb example, F(1000) would provide the total probability that the bulb fails within the first 1000 hours.

Given f(t) = 0.01 for 0 ≤ t ≤ 100.

1. Reliability R(t): R(t) = 1 - F(t) = 1 - ∫ 0.01 dt = 1 - 0.01t. Thus, R(t) = 1 - 0.01t for 0 ≤ t ≤ 100.
2. Hazard Rate λ(t): λ(t) = f(t) / R(t) = 0.01 / (1 - 0.01t) = 1 / (100 - t).
3. MTTF: MTTF = ∫ t * f(t) dt from 0 to 100 = ∫ 0.01t dt = [0.01 * t^2 / 2] from 0 to 100 = 0.005 * 10000 = 50 days.
4. Standard Deviation (σ): First, calculate E(t^2) = ∫ t^2 * f(t) dt = ∫ 0.01t^2 dt = [0.01 * t^3 / 3] from 0 to 100 = 10000 / 3 ≈ 3333.33. Variance Var(t) = E(t^2) - (MTTF)^2 = 3333.33 - 50^2 = 3333.33 - 2500 = 833.33. Standard deviation σ = √833.33 ≈ 28.87 days.
5. Median Time to Failure (tm): Set F(tm) = 0.5. Since F(t) = 0.01t, 0.01 * tm = 0.5, which gives tm = 50 days.