Outline Solutions to Tutorial Sheet 81. The exponential probability plot is :Exponential Probability Plot for Hrs to Fail99 ML EstimatesM ean: 1.41989795908070605030100 5 10Data1The plot gives the estimate of the mean as 1.41. Thus, since the mean is , the estimate of λ isλ1= 0.7091.41−0.709×1(i) p(X < 1) = F(1) = 1− e = 1− 0.4921 = 0.5079−0.709×2(ii) p(X > 2) = R(2) = e = 0.2422(iii) p(1.5 < X < 2.5) = F(2.5) − F(1.5) or−0.709×1.5 −0.709×2.5R(1.5) − R(2.5) = e − e = 0.3452 − 0.1699 = 0.1753The hazard function is h(t) = λ = 0.709 (i.e. constant) and the MTBF is 1.41 hours2. The Weibull plot shows a good fit. Estimates of α (scale) and β (shape) are given as 0.735 and2.61 respectively.For this value of β, (>2) we expect a hazard function shapePercentThis shows positive ageing. Failure rates are increasing at an increasing rate.β 2 .61 t t − − α 0 .735 R ( t ) = e = e (in units of 1 million hours)2.610.75 − 0.735 −1.054 Then R(0.75) = e = e = 0.3485 1 1 MTBF = αΓ1+ = 0.735Γ 1+ = 0.735Γ(1.38) = 0.888537 from tables of the gamma β 2.61 function. Thus the MTBF is approximately 889,000 hours.Weibull Probability Plot for Hrs(millions99 ML Estimates95 Shape: 2.608719080 Scale: 0.73496570605040302010 5 3 2 10.1 1.0Data3. The Distribution Overview Plot for the Part B’s exponential lifetimes is:Overview Plot for Part BNo censoringProbability Density Function ...