3: Let X₁. X, Exponential (8) (1) Find UMP test of size a for testing H₁ : 0 = 1 against H₁ : 0 > 1. (2) For n = 10 find UMP test of size a = 0.05 for testing H₂:0=1 against H₁:0 > 1.
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- In an effort to make the distribution of income more nearly equal, the government of a country passes a tax law that changes the Lorenz curve from y = 0.98x2.1for one year to y = 0.32x2 + 0.68x for the next year. Find the Gini coefficient of income for both years. (Round your answers to three decimal places.) after beforeAt time t = 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter λ. After the first birth, there are two individuals alive. The time until the first gives birth again is exponential with parameter λ, and similarly for the second individual. Therefore, the time until the next birth is the minimum of two exponential (λ) variables, which is exponential with parameter 2λ. Similarly, once the second birth has occurred, there are three individuals alive, so the time until the next birth is an exponential rv with parameter 3λ, and so on (the memoryless property of the exponential distribution is being used here). Suppose the process is observed until the sixth birth has occurred and the successive birth times are 25.8, 42.0, 52.0, 55.8, 59.2, 63.3 (from which you should calculate the times between successive births). Derive the mle of λ. [Hint: the likelihood is a…Find the absolute maximum and minimum values of ƒ(x) =10x (2 - ln x) on the interval [1, e2 ]
- Solve the initial value problems in Exercise for x as a functionof t.Find the root of the function f(x) = ln(x2) – 0.7 using bisection and false position methods. Use initial guesses of 0.5 and 2.0 with five iterations and five decimal places. Which method is more efficient in finding the root ? Why ?Show mathematically that MRS is increasing for U=1-ex +e2y and is diminishing for U=lnX +LnY
