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# Relaibility and Failure statisitcs and maths

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Please see attachment for PDF functions in question 1 - 3

1. A manufacturing company produces many different products for wholesale distribution. Its flagship product, a reverse cylindrical dipolar tube, has a failure distribution given by the following PDF:

f(t)= {█(0.000192t^2 0≤t≤25 years@0 elsewhere)┤

Find R(15) and MTTF

Find the probability that a failure will occur after the second year and by the end of the tenth year.

Find the median time to failure t_med.

Construct the hazard rate function, λ(t). Is it IFR, DRF, or CFR?

2. The time to failure in operating hours of a critical solid-state power unit has the hazard rate function

λ(t)=0.003(t/500)^0.5,for t≥0

What is the reliability if the power unit must operate unit must operate continuously for 50 hours?
Determine the design life if a reliability of 0.90 is desired.

3. A system contains 10 serially related identical components with CFR hazard λ_com. If a design life of 5 years with a reliability of 0.92 is required, what should the system MTTF and median time to failure be? What will the component MTTF be?

4. Time to failure for a product follows a Weibull distribution with β = 2 and η = 400 months.
What is the reliability at 150 months?
What is the probability of failure in 300 months?
Now suppose shape = 1. Find the reliability at 150 months. What is the reliability at 300 months, given that the product survived up to 150 months? Compare the two resulting values and comment on your observation

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Various questions and solutions involving reliability and failure analysis

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OK Here you go. I haven't had a chance to check through much of this work, arithmetical mistakes if any at all but I can of course do this tomorrow when I have more time, Thanks

A manufacturing company produces many different products for wholesale distribution. Its flagship product, a reverse cylindrical dipolar tube, has a failure distribution given by the following PDF:

f(t)= {█(0.000192t^2 0≤t≤25 years@0 elsewhere)┤

Find R(15) and MTTF

F(t)=∫_0^∞▒〖f(t)dt=∫_0^t▒〖0.000192t^2 dt+∫_t^∞▒0dt〗〗

F(t)=0.000192 t^3/3=0.000064t^3

R(t)=1-F(t)=1-0.000064t^3

R(15)=1-0.000064×15^3=0.784

MTTF=∫_0^∞▒〖t.f(t)dt=∫_0^25▒〖0.000192t^3 dt〗〗

MTTF=(_0^25)|0.000048t^4 | =18.75 yrs

Find the probability that a failure will occur after the second year and by the end of the tenth year.

From before the unreliability is derived from the function F(t)=0.000064t^3

So probability of failure after 2nd year is

F(2)=0.000064×2^3=0.000512

Probability of failure after 10 years is

F(10)=0.000064×〖10〗^3=0.064

Find the median time to failure t_med.

Median time to failure is when F(t)=0.5 so we can write

0.000064t^3=0.5
t=∛(0.5/0.000064)=19.84 yrs

Construct the hazard rate function, λ(t). Is it IFR, DRF, ...

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