Probability Density FunctionsExample #1Example #2Probability Density FunctionsThe area under the graph of the p.d.f. of a continuous random variable must be equal to one. In other words, over allpossible ( )Xxf x dx must be equal to one.The “integral of the function )(xfXover all possible x” is given by  00)()()( dxxfdxxfdxxfXXX where 00)(lim)( dxxfdxxfXbX and 00.)(lim)(aXaXdxxfdxxfNow, the area under the graph of the p.d.f. of a continuous random variable over the [a,b] corresponds to baXdxxfbXaP .)()(Also recall that the expected value of a discrete random variable is given byover allpossible ( ) * ( ).XxE X x f x . Similarly, the expected value of a continuous random variable is given by  .)(*)( dxxfxXEXExample #1The probability density function of X, the amount of time (in minutes) between arrivals/departures at the Phoenix Sky Harbor airport, is given by1.20 if 0( )1 if 01.2xXxf xe xSet up but do not evaluate an integral that could be used to verify that )(xfXis a valid p.d.f.Set up but do not evaluate an integral that corresponds to ).105( XPSet up but do not evaluate an integral that corresponds to E(X).Use Integrating.xls to verify that )(xfXis a valid p.d.f.Use Integrating.xls to find ).105( XPUse Integrating.xls to find E(X).Example #2The probability density function of a continuous random variable Y is given by1 if 2 12( )100 elsewhereYyf y Set up but do not evaluate an integral that could be used to verify that ( )Yf yis a valid p.d.f.Set up but do not evaluate an integral that corresponds to ( 9).P Y Set up but do not evaluate an integral that corresponds to E(Y).Use Integrating.xls to verify that ( )Yf yis a valid p.d.f.Use Integrating.xls to find ( 9).P Y Use Integrating.xls to find