# 1. Suppose the joint probability distribution function F(x.g) (actu...

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## Question 1
x = 0    x = 1    x = 2

y = 0    0.10      0.20      0.20

y = 1    0.20      0.20      0.10

Marginal Probability of y:

y         y = 0    y = 1

P(Y=y)    0.50      0.50

Expected Value of y, E(y) = 0*0.50 + 1*0.50 = 0.50

E(y^2) = 0^2*0.50 + 1^2*0.50 = 0.50

Var(y) = E(y^2) - E(y)^2 = 0.50 - 0.50^2 = 0.25

sd(y) = sqrt(var(y)) = 0.5

Marginal Probability of x:

x         x = 0    x = 1    x = 2

P(X=x)    0.30      0.40      0.30

Expected Value of x, E(x) = 0*0.30 + 1*0.40 + 2 *0.30 = 1

E(x^2) = 0^2*0.30 + 1^2*0.40 + 2^2*0.30 = 1.60

Var(x) = E(x^2) - E(x)^2 = 1.60 - 1^2 = 0.6

sd(x) = sqrt(var(x)) = 0.7745967

E(xy) = sum(y*f(x,y)*x) = 1*0.20*1 + 1*0.10 * 2 = 0.40

Cov(x,y) = E(xy) - E(x)*E(y) = 0.40 - 1*0.50 = -0.10

rho = Corr(x,y) = Cov(x,y)/sd(x)*sd(y) = -0.10/(0.7745967*0.5) = -0.2581989

### (a)

Under MSE loss, the best constant predictor of y is the population
mean ie. E(y) which is 0.50...

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