Suppose the chance of rain in Chicago and Minneapolis are correlated. Define X =

Question

Suppose the chance of rain in Chicago and Minneapolis are correlated. Define X = 1 if it is raining in Chicago some day and X = 0 if not; Similarly, define Y = 1 if it is raining in Minneapolis and Y = 0 otherwise. Now we know the probability of raining in Chicago and Minneapolis are both 0.35, i.e., P(X = 1) = 0.35 and P[Y = 1) = 0.35. Now (a) If given it rains in Chicago, the probability of rain in Minneapolis is 0.7. What is the correlation between X and Y? (b) If we know the correlation between X and Y is 0.5, what is the conditional probability that it rains in Minneapolis rains given it rains in Chicago?

Explanation / Answer

a) here mean of rain in chicago E(X)=0.35

and mean of Minnepolis E(Y) =0.35

similarly E(X2)=E(Y)2=0.35

hence Var(X)=Var(Y)=0.35-0.352 =0.2275

also E(XY)=0.7 as P(X=1,Y=1)=0.7*0.35=0.245

hence covariance(X,Y)=E(XY)-E(X)E(Y)=0.7-0.35*0.35=0.1225

therefore correlation =covar(X,Y)/(Var(X)*Var(Y))1/2 =0.5385

b)if correlation =0.5

hence Covar(X,Y)=0.5*0.2275=0.11375

hence E(X,Y)=Covar(X,Y)+E(X)*E(Y)=0.11375+0.35*0.35=0.23625

therefore conditional probability that if it rains in Minnepolis given it rains in chicago =0.23625/0.35 =0.675

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