(Cauchy dissemination.) An arbitrary variable X with thickness  co < X < co  co < b < 00 a>0 1 a f(x) = 1r a2 /7)2 is said to have a Cauchy dissemination with boundaries an and b. This dispersion is fascinating in that it gives an illustration of a constant irregular variable whose mean doesn't exist. Let a = 1 and b = 0 to acquire an exceptional instance of the Cauchy conveyance with thickness 1 f(x) = 1 + x= - co < X < CO Show that 1,1x1f(x) dx doesn't exist, hence showing that EIXJ doesn't exist. Clue: Compose 0, ir1 + x2 1 I 1 J-.1 + x.2O  x 1r 1 + x2 x  dx =Tr dx +r 0 dx and review that .Rdu/u)= In Sul. 

Let Xdenote the measure of time in hours that a battery on a sunlight based adding machine will work enough between openings to light adequate to re-energize the battery. Accept that the thickness for X is given by .f(x) = (5016)x-3 2

(b) Discover the articulation for the aggregate circulation work for X, and use it to discover the likelihood that an arbitrarily chosen sunlight based battery will last all things considered 4 hours prior to waiting be re-energized

(c) Figure out the normal time that a battery will last prior to waiting be re-energized

(d) Discover E[X2], and utilize this to discover the fluctuation of X.

Accept that the increment sought after for electric force in great many kilowatt hours throughout the following 2 years in a specific zone is an irregular variable whose thickness is given by KO= (1/64),0 0

(b) Discover the articulation for the total dispersion for X, and use it to discover the likelihood that the interest will be all things considered 2 million kilowatt hours.

(c) If the territory just has the ability to create extra 3 million kilowatt hours, what is the likelihood that request will surpass supply?

(d) Track down the normal expansion sought after.

The mass thickness of soil is characterized as the mass of dry solids per unit mass volume. A high mass thickness suggests a minimized soil with not many pores. Mass thickness is a significant factor in affecting root advancement, seedling development, and air circulation. Allow X to mean the mass thickness of Pima earth soil. Studies show that X is typically dispersed with p = 1.5 and a = .2 g/cm3. (a) What is the thickness for X? Sketch a chart of the thickness work. Show on this chart the likelihood that X lies somewhere in the range of 1.1 and 1.9. Discover this likelihood.

(b) Discover the likelihood that a haphazardly chosen test of Pima earth soil will have mass thickness under .9 g/cm3.

(c) Would you be shocked if an arbitrarily chosen test of this sort of soil has a mass thickness in overabundance of 2.0 g/cm3? Clarify, in view of the likelihood of this happening.

(d) What point has the property that lone 10% of the dirt examples have mass thickness this high or higher?

(e) What is the second creating capacity for .X?

A group of recent college grads decides to open a new airline company called BAir. They buy one airplane that has

seats. They estimate that two types of travelers will purchase tickets for a certain flight on a certain date:

Leisure travelers, who are willing to pay only the discounted fare $

Business travelers, who are willing to pay the full fare $ (where >).

After quite a bit of market research, they conclude that the number of leisure travelers requesting tickets for this flight will be greater than

n for sure, while the number of business travelers requesting tickets is random. Please assume that the leisure travelers always purchase their tickets before the business travelers do. (In practice, this is roughly true, which is why airfares increase as the flight date gets closer).

BAir wishes to sell as many seats as possible to business travelers since they are willing to pay more. However, since the number of such travelers is random and these customers arrive near the flight's departure date, a sensible strategy is for BAir to allocate a certain number of seats for full fares and the remainder, for discount fares.

The discount fares are sold first: The first customers requesting tickets will be charged $ per ticket and the remaining (at most ) customers will be offered the full price $ . Since leisure travelers are only willing to pay $ , they will decline to buy a full-fare ticket. Thus, if there are less than ticket requests from business travelers, some seats will not be sold (and BAir regrets not selling them to leisure travelers). Conversely, it is possible that some of the seats sold to leisure travelers for $ could have been sold to business travelers who would have been willing to pay $ .

Show that the problem of finding the optimal number of full-fare seats, , is equivalent to a newsvendor problem. Clearly define the underage cost, overage cost, and uncertainty. What is the critical ratio?

Suppose that demand for full-fare seats is normally distributed with a mean of 40 and a standard deviation of 18 (thus = 40 and = 18). There are =100seats on the flight and the fares are =$175 and =$400 . What is the optimal number of full-fare seats? (Fractional solutions are OK)

For each situation below, explain how the underage cost and the overage cost will change. How will this affect the optimal quantity reserved for full-fare customers? (No need to recalculate the optimal quantity - a qualitative answer is sufficient)

Situation 1: The full-fare tickets are fully refundable and, with some probability, each business traveler will cancel his or her ticket at the last minute, too late for BAir to re-sell the newly vacant seat.

Situation 2:Any unsold seats may be sold at the very last minute for a steeply discounted price, $ (<). These tickets are made available after the business travelers have requested tickets.