3A: Analyze Exponential and Logarith Analyze following function and graph it. f(x) = 3(-1) + 1 :(00,00) (1,000) Domain: Range: Asymptote: HA 9=34x XHA n_y=3x Original Function: f(x) = logs(x-1)-2 Describe Transformation: Domain: Range: End Behavior. Os Right (1) f(x) = (and as left Asymptote: End Behavior: xy Newpt 10 TO Original Function: 5 25 2 myz logo 52 Describe Transformation:
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- At 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…10. Suppose is a periodic function with period 11. What is the smallest period possible for the multivariate function and in what direction does it occur?Show mathematically that MRS is increasing for U=1-ex +e2y and is diminishing for U=lnX +LnY
- 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 beforeIn 1934, the Austrian biologist Ludwig von Bertalanffy derived andpublished the von Bertalanffy growth equation, which continues tobe widely used and is especially important in fisheries studies. LetL(t) denote the length of a fish at time t and assume L(0) = L0. Thevon Bertalanffy equation isdL/dt = k (A - L),where A = limtSqL(t) is the asymptotic length of the fish and k is aproportionality constant.Assume that L(t) is the length in meters of a shark of age t years.In addition, assume A = 3, L(0) = 0.5 m, and L(5) = 1.75 m. Solve the von Bertalanffy differential equation.Suppose that the Lorenz Curve for country A is given by yA = x 2.2 and for country B is yB = 0.3x 2 + 0.7x. Use the Gini index to determine which of the two countries has greater inequality in its income distribution.
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- The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as p(x) = −0.00022x3 + 0.014x2 − 0.0033x + 12.236 percent where x is the number of years since 2000, data from 0 ≤ x ≤ 50. 1. Determine the value of x in the domain 0 ≤ x ≤ 50 for which the percentage is predicted to be increasing most rapidly (Round your answer to three decimal places). 2. Thus, in which year is the percentage is predicted to be increasing most rapidly? 3. Calculate the percentage at that time. (Round your answer to two decimal places.) 4. Calculate the rate of change of the percentage at that time. (Round your answer to three decimal places.)Let x~ possion(alpha). Show that E[x(x-1)(x-2)....(x-k)]=ak+113.Find a sinusoidal function with a maximum value of 9 that occurs at x=4 , and a minimum value of -1 that occurs at x=11.