IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
X is uniform on (-1, 1) and Y = X^2. Which statement about X and Y is correct?
Covariance is zero even though Y is fully determined by X. By symmetry E[X] and E[X^3] are both zero, so Cov = E[X^3] - E[X]E[X^2] = 0. This shows uncorrelated variables need not be independent.
- ACov(X,Y) = 0 and X, Y are independent
- BCov(X,Y) > 0 because Y increases with |X|
- CCov(X,Y) < 0 because Y is a decreasing function of X on part of the range
- DCov(X,Y) = 0 although Y is completely determined by XCorrect
- Correlation is 1 because Y is a function of X
Explanation
Cov(X,X^2) = E[X^3] - E[X]E[X^2] = 0 - 0 = 0 by symmetry. Yet Y is a deterministic function of X, so they are dependent. Zero correlation does not imply independence.
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, Y) is bivariate normal with E[X]=50, E[Y]=100, SD(X)=10, SD(Y)=20 and correlation 0.6. What is E[Y | X = 60]?
- Given Θ = θ, the claim count N is Poisson(θ). Θ has a Gamma distribution with mean 2 and variance 1. What is Var(N)?
- Y = 3 - 2X, where X has variance 5. What is Cov(X, Y)?
- X is the number of claims in a year, with P(X=0)=0.6 and P(X=1)=0.4. Given X=0 the total claim cost Y is zero. Given X=1, Y is exponential w…
- X and Y are independent exponential random variables with rates 1 and 2 respectively. What is P(X < Y)?
- N is Bernoulli with P(N=1)=0.3. Given N=1, X has density 2e^(-2x) for x>0; given N=0, X has density e^(-x) for x>0. What is P(N=1 | X>1)?