**The amount of kerosene in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Suppose that the tank is not resupplied during the day so that x<=y, and assume that the joint density function of these variables is**

f(x,y) = 2, 0<x<y<1

f(x,y) = 0, elsewhere

a) Determine if X and Y are independent.

b) Find P(1/4 < X < 1/2 | Y = 3/4).

f(x,y) = 2, 0<x<y<1

f(x,y) = 0, elsewhere

a) Determine if X and Y are independent.

b) Find P(1/4 < X < 1/2 | Y = 3/4).

a)

\[g(x) = \int ^1 _{0}2 dy = 2\]

\[h(y) = \int ^y _{0}2 dx = 2y\]

\[g(x)h(y) = f(x,y)\], so they are independent.

b)

\[f(x|y) = \frac{f(x,y)}{h(y)} = \frac{1}{y}\]