mathematical problem

Suppose a weather-reporting station has an electronic wind speed monitor whose time to failure is known to follow an exponential model1θe−1/θ, 0 < θ < ∞.Let Y1 be the time until the monitor fails to work. The station also has a backup monitor; let Y2 be the time until this second monitor fails. Thus, the time Y until the monitoring is out of commission is the sum of two independent exponential random variables, Y1 and Y2. Thus Y =Y1 +Y2 hasagammapdf,sofY(y;θ)= 1ye−y/θ, 0≤y<∞, 0<θ<∞. θ2Five data points have been collected: 9.2, 5.6, 18.4, 12.1, and 10.7. Find the maximum likeli- hood estimate for θ.

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