Probability of joint pdf’s

Question

What’s the probability of P(X•Y<1) Given the pdf’s f_Y(y)=3/8•y² for 0<y<2

f_X(x)=3/8•x² for 0<x<2

its supposed to be around 0.08 but I just can’t find the right method

Answer 1

A suggestion: while you could immediately write the integration, why not draw a picture of the domain and range and plot x * y = 1

What you will see is, for x <= 1/2, y can range from 0 to 2.

For x > 1/2, y can range from 0 to 1/x.

Then, let's integrate in two parts.

For the first region, 0 <= x <= 1/2;0 <= y <= 2, we can take a short-cut, as we know that both f(x) and f(y) integrate to 1

∫0^1/2 3/8x² dx = 1/8x³ |0^1/2 = 1/8 * (1/2)³ = 1/64

For the second region, 1/2 <= x <= 2, 0 <= y <= 1/x

∫_1/2² 3/8x² ∫0^1/x3/8y² dydx = ∫_1/2² 3/8x² 1/8y³ |0^1/x dx=

∫_1/2² 3/8x² (1/(8x³) - 0) dx = ∫_1/2² 3/(64x) dx =

3/64 ln x |1/2² =

3/64 (ln 2 - ln 1/2) = 3/32 ln 2

1/64 + 3/32 ln 2 = .080608