Question

Let X and Y be continuous random variables with joint pdf fx.v (x, y)-3x, OSysx<1, and zero otherwise. a. b. c. d. e. What is the marginal pdf of X? What is the marginal pdf of Y? What is the expectation of X alone? What is the covariance of X and Y? What is the correlation of X and Y?

Answer 1

a)

marginal pdf of x

$\int_{0}^{x}$3x dy = 3x [ y]₀^x =3x² for 0≤x≤1

b)

marginal pdf of y

$\int_{y}^{1}$3x dx = 3/2[x²]¹_y = 3/2(1-y²) ,   for 0≤y≤1

c)

E[x] = $\int_{0}^{1}$x*3x² dx = 3/4[x^4]¹₀ = 3/4

d)

covariance =

Cov(X,Y) = E(XY)-E(X)E(Y)  
E(XY)=ΣXYP(XY)  
E(XY) = $\int_{0}^{1}\int_{0}^{x}$ xy*3x dy dx = $\int_{0}^{1}$3x²/2 [ y ]₀^x dx=  $\int_{0}^{1}$3x³/2 dx = 1/2

E[y] = $\int_{0}^{1}$ y*3/2(1-y²) dy = 3/4 - 1/2 = 1/4

so, Cov(X,Y) = E(XY)-E(X)E(Y) = 1/2-1/4*3/4 = 0.3125

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