Question

Let X,Y be random variables with the joint density function f

X,Y (x,y)={
cx^3y,0,0≤x≤1,0≤y≤2
0 Otherwise }
, where c is a constant. Find proper c, the marginal density function of Y and the correlation of X and Y.

Answer 1

The solution requires integrating the joint density function to find the constant c, the marginal density function of Y, and the correlation between the random variables X and Y by calculating respective expected values and variances.

Step-by-step explanation:

The student's question involves finding the appropriate constant c for a joint density function, calculating the marginal density function of Y, and determining the correlation between two random variables X and Y with given densities. To find c, we integrate the joint density function over the given ranges for X and Y:

∫₀¹∫₀²₀ c x³ y dy dx = 1

Once c is found, the marginal density function of Y is determined by integrating the joint density function over the range of X:

f_Y(y) = ∫₀¹ f_{X,Y}(x,y) dx

To find the correlation, we use the formula that involves the expected values of X, Y, and XY. These expected values are determined by integrating the respective moments using the joint density function. Note that the correlation requires that both X and Y have finite variances.