1. Average Value of a Continuous Function. MATH 1003 Calculus and Linear Algebra (Lecture 30) Average Value of a Continuous Function

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1 1. Average Value of a Continuous Function MATH 1 Calculus and Linear Algebra (Lecture ) Maosheng Xiong Department of Mathematics, HKUST Definition Let f (x) be a continuous function on [a, b]. The average value of f (x) on [a, b] is given by 1 b f (x). b a a Interpretation One may divide the interval [a, b] into n sub-intervals. Then from each subinterval I k, we sample one point x k and take the average: Ā n f (x 1) + + f (x n ) n (f (x 1 ) + + f (x n )) b a 1 }{{ n } b a. Riemman sum Therefore, lim n Ā n 1 b b a a f (x) Average Value of a Continuous Function Given the demand function p D(x) 1e.5x, find the average price (in dollars) over the demand interval [4, 6]. Solution The average price is b p D(x) 1e.5x b a a ( ) e.5x 6 1 ( e e ) $ Probability Density Functions A probability density function must satisfy the following three conditions: 1. f (x) for all real x;. The area under the graph of f (x) over the interval (, ) is exactly 1;. If [c, d] is a sub-interval of (, ), then the probability of x falling in [c, d] is defined by: P(c x d) d c f (x)

2 A Mini- Suppose there are two stocks A and B for investment. Historical data have shown the following information. Stock A is now $6 per share, and in one year its share price increase x satisfies a probability density distribution { ( f A (x) x x ) : 5 x 15, : for x elsewhere. Stock B is now $7 per share, and in one year its share price increase x satisfies a probability density distribution: A Mini- (a) Find the probabilities of a loss investing in A and B in one year. (b) Find the probabilities of a 1% gain investing in each of the stocks A and B. f B (x) { 5 ( 9 + 8x x ) : 1 x 9 : for x elsewhere. (a) Find the probabilities of a loss investing in A and B in one year. A Mini- A Mini- Solution for (a) For A it can be checked that f A (x) and 5 5 f A(x) 1. Investment loss means x. (75 + 1x x ) P A (x ) 5 4 (75x + 5x x ) Interpretation The chance of loss investing in A in one year is 15.6%. Solution for (a) For B it can be checked that f B (x) and 9 1 f B(x) 1. Investment loss means x. (9 + 8x x ) P B (x ) 1 5 (9x + 4x x ) Interpretation The chance of loss investing in B in one year is.8%. Which one is more risky?

3 A Mini- A Mini- Solution for (b) 1% profit from A means x (75 + 1x x ) P A (x 6) 6 4 (75x + 5x x ) Interpretation The chance of 1% gain investing in A in one year is 4.5%. Solution for (b) 1% profit from B means x (9 + 8x x ) P B (x 7) 7 5 (9x + 4x x ) Interpretation The chance of 1% gain investing in B in one year is 1.4%. Combination of A and B - Portfolios The life expectancy (in years) of a microwave oven is a continuous random variable with probability density function { x f (x) (x+) otherwise (a) Find the probability that a randomly selected microwave oven lasts at most 6 years. (b) Find the probability that a randomly selected microwave oven lasts 6 to 1 years. Solution (a) Let Y be the life expectancy of a microwave oven. Then Pr(Y 6) 6 6 (x + ) x + (x + )

4 Income Distribution of a Society Solution (b) Pr(6 Y 1) 6 (x + ) 1 x The following is the family income distribution in U.S., 6 Income Level x y < $,.. < $8,.4.1 < $6,.6.7 < $97,.8.49 The variable x represents the cumulative percentage of families at or below a given income level, and y represents the cumulative percentage of total family income received. For example, the point (.4,.1) means that the bottom 4% of families (incomes under $ 8,) received 1% of the total income for all families. Income Distribution of a Society Absolute Equality and Inequality The blue curve in the graph of y f (x) is called the Lorenz curve. For example, it may be expressed by with α 1. f (x) x α The variable x represents the cumulative percentage of families at or below a given income level, and y represents the cumulative percentage of total family income received. For example, (.4,.1) is the point on the Lorenz curve and means that the bottom 4% of families received 1% of the total income for all families. Remarks Any Lorenz curve is below the 45 degree line. If the income were distributed with absolute equality, the Lorenz curve would coincide with the 45 degree line. If the income were distributed with absolute inequality, the Lorenz curve would coincide with the horizontal axis and the right vertical axis.

5 Gini Index Let us define the Gini index, a measurement of the degree of inequality in the distribution of income in a society: Definition If the Lorenz curve is given by y f (x), then Gini index (G) is defined to be G Remarks (x f (x)) G is the ratio between A 1 (the area enclosed by y x and y f (x)) and A (the area enclosed by y x, y and x 1) and G 1. As G increases, the degree of inequality in the distribution of income increases. Gini Index A country is planning changes in tax structure in order to provide a more equitable distribution of income. The two Lorenz curves are: f (x) x. currently and g(x).4x +.6x proposed. Will the proposed changes work? Gini Index Consumers Surplus Solution Currently, the Gini index is ( x (x x. ) x. ) 1..9 After the proposed changes, the Gini index is (.6x [x (.4x +.6x )].6x ) 1. Therefore, the proposed changes will work because the Gini index becomes lower. If ( x, p) is a point on the graph of the price-demand equation p D(x), the consumers surplus CS at a price level of p is CS( x) x D(x) p. This is the area between p p and p D(x) from x to x x. The consumers surplus represents the total savings to consumers who are willing to pay more than p for the product but are still able to buy the product for p.

6 Consumers Surplus Consumers Surplus Find the consumers surplus at a price level of p 1 for the price-demand equation p D(x).x. Solution First, find the demand when the price is p 1: 1 p. x x 4. Solution Second, find the consumers surplus: CS( x) x 4 4 D(x) x.x 1 8.x (8x.1x ) 4 16 $16.