Review Problem on Con tinuous Random VariablesThe bid that a competitor makes on a real estate property is estimated tobe somewhere between 0 and 3 million dollars. Specifically, the bit X is viewedto be a continuous random variable with density function:f(x)=c¡9 − x2¢for 0 <x<3=0otherwise.You make a bid (without knowing the competitor’s bid). The higher of thetwo bids win.Questions1. Find the value of c that makes f(x) a legitimate density function?2. Find the cumulative distribution function, F(x). Use the cum ulative dis-tribution to determine the probability that you lose the bid if you make abid of 2 million? 1 million?3. Find the expected value and standard deviation for the competitor’s bid.What is the probability that the competitor’s bid is within one standarddeviation of the mean?4. How much should you bid so that you have a 90% chance of winning?Answ ers1. cR30¡9 − x2¢dx =1. This implies ch9x −x33i30=1. Hence c [27 − 9] = 1.So c =1182. F (x)=xZ−∞f(x)dx =Rx0(9−x2)18dx =x2−x354for 0 <x<3.F(x)=0 for x < 0 and 1 for x > 3.Hence P (X>2) = 1 − F (2) = 1 −¡22−854¢=854P (X>1) = 1 − F (1) = 1 −¡12−154¢=28543. E(X)=∞Z−∞xf(x)dx =3Z0x(9−x2)18dx =hx24−x472i30=98E(X2)=∞Z−∞x2f(x)dx =3Z0x2(9−x2)18dx =hx36−x590i30=4.5 − 2.7=1.81V (X)=1.8 −¡98¢2= .534375.So the standard deviation is√.534375 =.731Finally, one standard deviation of the mean is 1.125 − .731 = .394 to1.125 + .731 = 1.856.F (1.856) − F (.394) =hx2−x354i1.856.394= .8096 − .1959 = .6137.4. We want the value of x so that F (x)=.9Since F(x) is an increasing function we can find this b y trial and error.The value of x is approximately