The weekly demand for propane gas (in 1000s of gallons) from a particular facility is an rv ?X ?with pdf a.? ?Compute the cdf of ?X?. b.? ?Obtain an expression for the (100?p?)th percentile. What is the value of ? c.? ?Compute ?E?? ?) and ?? ?? . d.? ?If 1.5 thousand gallons are in stock at the beginning of the week and no new supply is due in during the week, how much of the 1.5 thousand gallons is expected to be left at the end of the week? [?Hint?: h(x)Let amount left when demand = ?x.? ]

Solution Step 1: It is given that the weekly demand for propane gas (in 1000s of gallons) from a particular facility is an rv X with pdf We have to find the cdf X 100th percentile , median , mean and variance. Step 2 : a) We have to find the cumulative distribution function. x F(x) = P(X x) = f(x)dx 1 x = 2( 1 1/x ) dx 1 = 2 [x+1/x] x 1 = 2[x+1/x - 2] Also note that F(1)=0, F(2)=1. So we can express the cdf of X in the form. b) we have to obtain an expression for 100th percentile. And also want to find the median . In order to find the percentile set F(x)= P , then find the value for x. In 100th percentile the value of P will be = 1 So here it will be F(x)= 2( x+1/x - 2) = 1 2 = (x + 1- 2x) = x/2 2 = (x - 2x-x/2 + 1)=0 = (x-2) (x-½) = 0 = x= 2 or x= ½ So the 100th percentile will be at x= 2 or x=1/2 . Similarly The median is the 50th percentile, which means here P= 0.5 So F(x)= 2( x+1/x - 2) = 0.5 = (x + 1- 2x) = x/4 = (x - 2x-x/4 + 1)=0 2 = (x - 9/4 x +1) = 0 By using quadratic formula b± b 4ac x= 2a X = 9/8 ± 17/8 1