Mean of a random variable Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table gives the joint distribution for X and Y .!enter image description here Can someone help us understand this chart...

This is the joint density. To find $\mu_X$ and $\mu_Y$, you need the marginal densities. That is, you need the density of just $X$ and the density of just $Y$, each of them by themselves. We can get this quickly from the table by adding up each row and column.

For example, the probability that $X = 1$ and $Y = 1$ is $0.10$. The probability of $X = 1$ and $Y = 2$ is $0.05$. The probability of $X = 1$ and $Y = 3$ is $0.02$. These are the only ways that $X$ can be $1$. So, the probability that $X = 1$ is the sum, $0.1 + 0.05 + 0.02 = 0.17$. Adding up the second row gives the probability that $X = 2$ and the third row the probability that $X = 3$. Then, you have the density of $X$ and you find the expected value just as you would if $Y$ were never in the problem. Similarly, add up the columns to get the marginal density of $Y$ and then calculate $\mu_Y$ from that.