计算R中联合概率质量函数的协方差

Question

I have a joint probability mass function of two variables X,Y like here

How can I calculate the covariance in R?

I created two vectors x,y and fed them into cov(), but I get the wrong result.

How can I do this right?

Thanks in advance and happy coding!

Answer 1

Since SO is a coding forum, I'll leave working out the math/stats details up to you. Here is an implementation in R.

1. We start by noting the sample spaces for X and Y

    # For G G <- 0:3; # For R R <- 0:2; 

2. The joint probability mass function is given by the following matrix

    joint_pmf <- matrix( c(4/84, 12/84, 4/84, 18/84, 24/84, 3/84, 12/84, 6/84, 0, 1/84, 0, 0), ncol = 3, byrow = T); 

3. We calculate the population means

    # For G mu_G <- rowSums(joint_pmf) %*% G; # For R mu_R <- colSums(joint_pmf) %*% R; 

4. We can make use of the theorem Cov(X, Y) = E[XY] - E[X]E[Y] to calculate the covariance

    cov_GR <- G %*% joint_pmf %*% R - mu_G * mu_R; # [,1] #[1,] -0.1666667 

   where we have used the fact that E[G] = mu_G and E[R] = mu_R are the respective population means.