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