Question

Let X and Y be two jointly continuous random variables with joint pdf f_xy (x, y) = 10x²y; 0 ≤ y ≤ x ≤ 1. The marginal density f_γ (y) is:
1. \(\dfrac{10}{3}y^3(1-y);0 \le y \le1\)
2. \(\dfrac{10}{3}y(1-y^2);0 \le y \le1\)
3. \(\dfrac{10}{3}y^2(1-y^2);0 \le y \le1\)
4. \(\dfrac{10}{3}y(1-y^3);0 \le y \le1\)

Answer 1

Correct Answer - Option 4 : \(\dfrac{10}{3}y(1-y^3);0 \le y \le1\)

Given

F_xy(x, y) = 10x²y, 0 ≤ y ≤ x ≤ 1

Concept used

Marginal probability is the probability of an event irrespective of the outcome of another variable.

Calculation

Marginal density f_y(y) = \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqGHRiI8ieWacaWFMbWaaeWaa8aabaWdbiaa-HhacaGGSaGaa8hO % aiaa-LhaaiaawIcacaGLPaaacaWFKbGaa8hEaaaa!401C! \smallint f\left( {x,\;y} \right)dx\)

⇒ \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaqfWaqabSWdaeaaieWapeGaa8hEaiabg2da9iaa-Lhaa8aabaWd % biaaigdaa0WdaeaapeGaey4kIipaaaaa!3C61! \mathop \smallint \nolimits_{x = y}^1 \)(10x²ydx

After integration we get

⇒ 10y(x³/3)¹_{x = y}

⇒ (10y/3)(1 - y)³

∴ The marginal density fγ (y) is (10y/3)(1 - y)3