Question

1 * Consider the following joint distribution for the weather in two consecutive days. Let X and Y be the random variables for the weather in the first and the second days, whereas the weather is coded as 0 for sunny, 1 for cloudy, and 2 for rainy. 0 0.2 0.2 0.2 10.1 0.1 0.1 2 0 0.1 0 (a) Find the marginal probability mass functions for X and Y (b) Are the weather in two consecutive days independent? (c) Calculate the expectation and variance for X and Y (d) Calculate the covariance and correlation between X and Y. Are they correlated?

Answer 1

below is joint probability distribution of X and Y:

	X
Y	0	1	2	Total
0	0.2000	0.2000	0.2000	0.6000
1	0.1000	0.1000	0.1000	0.3000
2	0.0000	0.1000	0.0000	0.1000
Total	0.3000	0.4000	0.3000	1.0000

a)

marginal distribution of X:

X	P(X)	XP(X)	X^2P(X)
0	0.3000	0.0000	0.0000
1	0.4000	0.4000	0.4000
2	0.3000	0.6000	1.2000
total	1.0000	1.0000	1.6000
E(X)	=		1.0000
E(X^2)	=		1.6000
Var(X)	E(X^2)-(E(X))^2		0.6000

marginal distribution of Y:

Y	P(Y)	YP(Y)	Y^2P(Y)
0	0.6000	0.0000	0.0000
1	0.3000	0.3000	0.3000
2	0.1000	0.2000	0.4000
total	1	0.5	0.7
E(Y)	=		0.5000
E(Y^2)	=		0.7000
Var(Y)	E(Y^2)-(E(Y))^2		0.4500

b)

here as P(X=0)*P(Y=0)=0.3*0.6 =0.18 ; which is not equal to P(X=0,Y=0)=0.20 ; hence X and Y are not independent

c)

expectation of X =E(X)=1

expectation of Y =E(Y)=0.5

Variance of X =Var(X)=0.6

Variance of Y =Var(Y)=0.45

d)

here E(XY)

= $\tiny \sum$ xyP(x,y)=0*0*0.2+1*0*0.2+2*0*0.2+0*1*0.1+1*1*0.1+2*1*0.1+0*2*0+1*2*0.1+2*2*0=0.5

hence Covariance =Cov(X,Y)=E(XY)-E(X)E(Y)=0.0000

therefore Correlation =Cov(X,Y)/(Var(X)*Var(Y))^1/2 =0

as correlation between X and Y is 0 ; threfore X and Y are uncorrelated