(a) By a process of repeated sampling from a stationary population, the probability density function g(x) of the population age-distribution was derived for all x>0. (i) By implementing and using g(x), construct an expression for the mean lifetime for members of this population. (ii) Detail and formulate how the median lifetime of members of this population can be determined using g(x).
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- What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?Suppose X1,...,Xn are measurements of the mass of an object. The measurements are normally distributed, with mean equal to mu , the mass of the object, and standard deviation equal to 1 microgram. Find (a) The likelihood function. (b) The log-likelihood function (c) the maximum likelihood estimator for the unknown parameter, andThe operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 100 cfs (cubic feet per second).a) Find the probability that the demand will exceed 200 cfs during the early afternoon on arandomly selected day.b) What water-pumping capacity should the station maintain during early afternoons sothat the probability that demand will exceed capacity on a randomly selected day isonly .01?
- Suppose X has a lognormal distribution with parameters mu = 1.5 and sigma = 0.2 . Find the 33rd percentile?An investment of $100 produces rate of return as follows In year 1: a gain of 10 percent In year 2: a loss of percent In year 3: a loss of 8 percent In year 4: a gain of 3 percent. Calculate the value of the investment at the end of the fourth year and calculate the mean annual rate of return.Consider the following (actual real-world) data of total cumulative coronavirus cases diagnosed in the United States by the following days: Sunday, 3/15 -- 3613 Monday, 3/16 -- 4596 Tuesday, 3/17 -- 6344 Wednesday, 3/18 -- 9197 Thursday, 3/19 -- 13779 Friday, 3/20 -- 19367 Saturday, 3/21 - 24192 Sunday, 3/22 - 33592(a) Does this data suggest exponential growth in the total number of cases diagnosed as function of the number of days that have elapsed? Explain how you would use the data to support your conclusion. (b) Let P(t) denote the total number of people in the US who have tested positive for the coronavirus on or before t days after March 15. Let P0=3613, the total number of people in the US who had tested positive for coronavirus by March 15. Then if we use an exponential growth model, P(t)=P0ekt, how would you use the above data to estimate the value of k? (c) Based on this data alone , how many total coronavirus cases would you expect to be diagnosed in the United States by…
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- The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 1000 cfs (cubic feet per second). a)Of the three randomly selected afternoons, what is the probability that on at least two afternoons the demand will exceed 700 cfs?A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.Let X|Λ have an exponential distribution with conditional survival function SX|Λ(x|λ) = e−λx. Let Λ have a Gamma distribution with parameters α and θ. Find the unconditional or marginal survival function of X.