In some applications, a sample of failure times comes from a mixture of subpopul
ID: 3182621 • Letter: I
Question
In some applications, a sample of failure times comes from a mixture of subpopulations. (a) Write down the expression for the CDF, PDF, and reliability function for a mixture of two exponential distributions with rates lambda_1 and lambda_2 (subpopulations 1 and 2. respectively) and p being the proportion from subpopulation 1. lambda_1 notequalto lambda_2. (b) What is the shape of the hazard rate function and prove that the mixture of two exponential distributions with different rate parameters always has that shape. (c) Find the MTTF of the mixture of two exponential distributions.Explanation / Answer
a.) CDF and PDF of exponention is 1 ex and ex respectively. Using M.G.F we can easing proff that mixture of two exponential distribution with 1 and 2 has PDF and CDF of (1+ 2) e(1+2)x and 1 e(1 + 2)x..
b.) The shape of the hazard function id s constant and whatever be the value pf lambda the shape of the hazard function remains the same always, since we have seen that a change in the parameters of the mixture only changes the lambda parameter and not the form of distribution the hazard function shape always remains the same. This can also be seen by plotting a graph of the hazard function with different lambda values.
c.) Since MTTF ids not a standard statisstical abrebriation, I can not answer question related to that.
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