Suppose X has the pdf given by and U = X and V = X^2. Find Cov(U, V) Are U and V

Question

Suppose X has the pdf given by and U = X and V = X^2. Find Cov(U, V) Are U and V independent? Justify your answer. Suppose that the daily demand for gas (in thousands of litres) at a certain gas station is an exponential random variable with mean equal to 2. Every morning the station's tank is filled to capacity. What is the probability that they sell more than 2000 litres in a given day? Find the median daily sales for the gas station in thousands of litres. That is, the value m such that the gas station sells less than in thousand litres in a day 50% of the time and sells more than m thousand litres in a day the other 50% of the time. What size must the tank be to ensure that the probability of running out of gas on any given day is only 1%

Explanation / Answer

(2)

(a)

P(Xx) = 1-^(-.x) where =½ (reciprocal of the mean)
So P(X2) = 1-(-.2) giving P(X>2) = 1-[1-(-.2)] = ^(-1½) = 0.223 = 22.3%

(b)

The median sale = the mean or expected sale = 2000 litres/day (given).Thus m=2
(c)

P(Xx) = 99% = 0.99 = 1-^(-½.x) to be solved for x:
^(-½.x) = 0.01 Take natural logs of each side : Ln(0.01) = -½.x
x = 9.21 so capacity needed = 9210 litres.

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