Question

From the following bivariate probability distribution, find the coefficient of correlation.

Answer 1

The coefficient of correlation is :

Here Var(x) = E(x²) – [E(x)]²

‘To calculate, obtain the marginal probility distribution of x:

Here : 0.05 + 0.25 + 0.1 

= 0.4 and 2 : 0.4 + 0 + 0.2 = 0.6

E(x) = Ex . P(x) = 1 × 0.4 + 2 × 0.6 = 1.6

E(x²) = Ex² . P(x) = 1² × 0.4 + 2² × 0.06 = 2.8

∴ Var (x)= 2.8 — (1.6)² = 0.24

The marginal probability distribution of y:

E(y) = Ey P(y) = (-1) × 0.45 + (-2) × 0.25 + 0 × 0.3 = – 0.95

E(y²) = (-1 )² × 0.45 + (-2)² × 0.25 + 0² × 0.3 

= 0.45 + 1 + 0 = 1.45

∴ Var(y) = E(y²) – [E(y)]² 

= 1.45 – (-0.95)² = 1.45 – 0.9025 = 0.5475

cov (x, y) = E(xy) – E(x) . E(y)

= (1 × (-1) × 0.05) + (2 × (-1) × 0.4) + (1 × (-2) × 0.25) + (2 × (-2) × 0) + (1 × 0 × 0.1) +(2 × 0 + 0.2)

= -0.05 -0.8 -0.5 + 0 + 0 + 0

E(xy) =-1.35

∴ cov(x,y) =-1.35- 1.6 x (-0.95) 

= -1.35 + 1.52 = -0.17