1. SupposeX1,..., Xniid∼Exponential(λ) is a set of n observations drawn independently from an Exponential distribution.   (d) Set the score function equal to zero and solve for the parameterλ. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.

Here's a table with average monthly price of a stock from Jan 2021-July 2021. Month Price Jan 29000 Feb 27000 Mar 31000 Apr 30000 May 32000 June 28000 July 29000 Using a 3-period weighted moving average with weights (wt, wt-1, W-2) = (0.6, 0.2, 0.2), the predicted price August is Using an exponential smoothing model with = 0.4, and with the initial predicted price for Jan the same as the observed price for January, the predicted price for August is % Now, let's suppose that we finally observe that the price for August is 30000. The absolute percent error (APE) of your prediction for August is using the weighted moving average method and % using the exponential smoothing model. (Note: Round your answer to the nearest 2 decimal places. For your APE answer for example let's say you get 0.06786 from Excel, then you should type-in 6.79 as your answer here since it is represented as percentage. Do not enter any symbols other than integers and decimal.)

A study done in 1994, of Maryland's portion of Chesapeake Bay found the following information about stocks of market-sized oysters.  The stock population was 218 million, the stock will not exceed 5089 million oysters, and the intrinsic exponential growth rate is 0.274 per year.

A individual, with a utility function lnW and an initial wealth level W0, is faced with a fair gamble of winning or losing $h (where W0 > h > 0) with 50-50 chance. (a) Is this individual risk averse? Explain.(b) Suppose that the individual is willing to pay up to an amount of f in order to avoid such a gamble. Give the equation that determines f, and solve the equation for f (i.e., express f in terms of W0 and h). (c) Show that f increases as h increases.

4. You point your Geiger counter at a banana, waiting for it to click. You know that the time you have to wait is exponentially distributed with rate >= 1 Unfortunately, 10% of all Geiger counters have been sabotaged by Big Banana. A sabotaged Geiger counter, when pointed at a banana, will instead click after every 60 seconds to lull you into complacency. (a) Determine the CDF of the waiting time until your Geiger counter clicks (you do not know if your Geiger counter has been sabotaged). (b) If you have been waiting for 30 seconds and your Geiger counter still hasn't clicked yet, what is the probability that it has been sabotaged?