IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
The joint density of X and Y is f(x,y) = x + y for 0 < x < 1 and 0 < y < 1. What is the marginal density of X for 0 < x < 1?
The marginal density of X is x + 1/2 on (0,1), obtained by integrating the joint density over y from 0 to 1; it integrates to 1.
- Ax + 1
- Bx + 1/2Correct
- C2x
- Dx + y
- x/2 + 1
Explanation
The marginal density of X is found by integrating out y: the integral of (x + y) dy from 0 to 1 equals x + 1/2. Check: the integral of x + 1/2 over (0,1) is 1/2 + 1/2 = 1. Option x + 1 forgets the factor 1/2 from integrating y.
Did you get it right without looking?
One question tells you little. A timed set on Jointly distributed random variables shows your real accuracy, how long you take and where you lose marks.
More Jointly distributed random variables questions
- X and Y have joint density f(x,y) = 2 for 0 < x < y < 1 (zero elsewhere). What is the marginal density of X for 0 < x < 1?
- Given Θ=θ, N is Poisson with mean θ. Θ is exponential with mean 2. Which is the value of Var(N)?
- X is a Bernoulli indicator with P(X=1)=0.5. Given X=0, Y is uniform on (0,10); given X=1, Y is uniform on (0,20). What is P(X=1 | Y=15)?
- Y = 3 - 2X, where X has variance 5. What is Cov(X, Y)?
- For random variables X and Y, E[X] = 4, E[Y] = 5 and E[XY] = 23. What is Cov(X, Y)?
- X is uniform on (-1, 1) and Y = X^2. Which statement about X and Y is correct?