Let .S; P / be a sample space and let A S be an event. Define a random variable IA whose value at s 2 S is IA.s/ D ( 1 if s 2 A, and 0 otherwise. The random variable IA is called an indicator random variable because its value indicates whether or not an event occurred. Prove: E.IA/ D P .A/.

Reece Witcher 2/27/18 Improper Integrals ● Improper integrals are integrals who have a bound that goes to infinity or when an integral has an asymptote at a bound ○ If there is an asymptote, you should separate the integral into two other integral and substitute the asymptote value with a limit on the appropriate side ○ ● The infinite bound can be pull out by substituting it for a variable and taking the limit of the variable as it goes to infinity 2 ○ lim a →infinitydx a ● You should then complete the integration ○ NOTE: make sure to use t with the solution ● Plug in the bounds, then plug in the limit ● If the bound value is infinite or undefined, the integral is divergent ● If the value is finite (a normal number like 1 or 800), the number is convergent ● Tips: ○ This strategy is useful for solving problems with asymptotes ■ Such as 1/x1, which has an asymptotes at x=1 ○ Don’t forget to solve for the bounds when you solve an integral using usub Arc Length b ● The formula for arc length is ∫a√❑ ● If you are given the f(x) formula, simply take the derivative and plug it into the equation ● Tips: ○ This formula is can be used to solve for the length along any curve ○ If you are calculating a shape that is symmetrical, simply set the bounds at