Query - Variance

[Avishek]

Given ->
Y = 1 Y = 2 Y = 3
X = 1 0.05 0.15 0.20
X = 2 0.15 0.15 0.30

Question -> The variance of Y is closest to?

Soln:

Firstly I have done p(Y=1) = .05+.15=.20; p(y=2) = .15+.15=.30; p(y=3) = .20+.30=.50
Then have calculate the mean μ, i.e. .20(1)+.15(2)+.30(3) = 2.3

Advise me as to what should be my next step?

Thanks, Avishek

 

[David Harper CFA FRM]

Avishek,

I have a bit of trouble interpreting the use of two X series. However, if I assume your sentence about probabilities, then:

Var(y) = Covariance (y,y) = E(yy) - E(y)E(y) = E(y^2) - E(y)^2

E(y) = Average(y) = (20%)(1)+(30%)(2) + (50%)(3) = 2.3
E(y^2) = Average (y^2) = (20%)(1^2)+(30%)(2^2)+(50%)(3^2)= 5.9

E(y^2) - E(y) = 5.9 - 2.3^2 = 0.61 is variance of y. Stand dev (y) SQRT(0.61) = 0.781.

David