Question

Consider a box containing 25% white, 50% black and 25% blue balls. Extract with replacement 2 balls. Denote by X and Y the random variables representing the number of white and, respectively, black balls obtained. Find the distribution of XY and calculate ρ(X, Y ).

Answer 1

sample (X,Y) probability

2,0 $0.25\cdot0.25=0.0625$

1,1 $0.25\cdot0.5=0.125$

0,2 $0.5\cdot0.5=0.25$

0,0 $0.25\cdot0.25=0.0625$

correlation coefficient:

$\rho (X,Y)=\frac{Σ(x_i - x̄)(y_i - ȳ)}{√(Σ(x_i - x̄)^2Σ(y_i - ȳ)^2 }=-0.4545$