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Recall-Project Assumptions Assumption 1. The same 19 companies will each bid on future similar leases only bidders for the tracts(This assumption is important for the Nash concept) Assumption 2. The geologists employed by companies equally expert (evidence- the Mean of errors of all historical leases is 0 ) on average, they can estimate the correct values of leases. evidence each signal for the value of an undeveloped tract is an observation of a continuous random variable, S v,

Recall-Project Assumptions Assumption 3. Except for their means, the distributions of the S v ’s are all identical (The shape /The Spread)-This allows us to treat all the 20 historical leases as one sample -> We use the sample to show that mean of errors is 0 & find the standard deviation of errors) Assumption 4. All of the companies act in their own best interests, have the same profit margins, and have the same needs for business. Thus, the fair value of a lease is the same for all 19 companies.

4 Integration, Integrals 2. INTEGRALS What would happen if we computed midpoint sums for a function which might assume negative values in the interval [a, b]? Where f(m i ) < 0, the product f(m i )  x is also negative. Thus, the midpoint sums will approximate the “signed area” of the region between the x-axis and the graph of f, over [a, b]. This is the algebraic sum of the area above the axis, minus the area below the axis. (material continues) Integration. Integrals a b +   ICT

5 Integration, Integrals the integral of f over [a, b] is and it represents the algebraic sum of the signed areas of the regions between the horizontal axis and the graph of f, over [a, b].

6 Integration Applications Fundamental Theorem of Calculus - Example : applies to p.d.f.’s and c.d.f.’s Recall from Math 115a Fundamental Theorem of Calculus. For many of the functions, f, which occur in business applications, the derivative of with respect to x, is f(x). This holds for any number a and any x, such that the closed interval between a and x is in the domain of f.

7 Integration, Calculus the inverse connection between integration and differentiation is called the Fundamental Theorem of Calculus. Fundamental Theorem of Calculus. For many of the functions, f, which occur in business applications, the derivative of with respect to x, is f(x). This holds for any number a and any x, such that the closed interval between a and x is in the domain of f. Example 7. Let f(u) = 2 for all values of u. If x  1, then integral of f from 1 to x is the area of the region over the interval [1, x], between the u-axis and the graph of f.

8 Integration, Calculus The region whose area is represented by the integral is rectangular, with height 2 and width x  1. Hence, its area is 2  (x  1) = 2  x  2, and (1, 2)(x, 2) x 2 x  1 In the section Properties and Applications of Differentiation, we saw that the derivative of f(x) = m  x + b is equal to m, for all values of x. Thus, the derivative of with respect to x, is equal to 2. As predicted by the Fundamental Theorem of Calculus, this is also the value of f(x). The next example uses the definition of a derivative as the limit of difference quotients.