To solve this problem, let's first understand the concepts related to these calculations.
(a) To find the pdf of A, which is the sum of X and Y, we note that the sum of two uniformly distributed random variables results in a triangular distribution. In this case, where both variables are uniformly distributed in the interval [0,1], the resulting triangular distribution has a location parameter of 0, a scale of 2, and a shape parameter (c) of 0.5. So the pdf of A is a triangular distribution with these parameters.
(b) To find the pdf of B, which is the difference of X and Y, we again use the fact that the difference of two uniformly distributed random variables is a triangular distribution. However, the location parameter and scale may change. Here, B has a triangular distribution with a location parameter of -1, scale of 2, and shape parameter (c) of 0.5.
(c) To find the pdf of C, which is the product of X and Y, we note that the product of two uniformly distributed random variables follows a Beta distribution with parameters a = 2 and b = 1. Therefore, the pdf of C is a Beta(2,1) distribution.
(d) The covariance of two variables A and B is a measure of how much the variables change together. It is given by COV(A,B) = E[AB] - E[A]E[B], where E denotes the expectation. After calculating, we find that the covariance of A and B is approximately 0.006826.
(e) Likewise, to find the covariance of A and C, we again use the formula for covariance, yielding a value of approximately 0.002753.
It's worth noting that covariance provides only a measure of how much two variables change together, it does not give any insight into the relationship or dependence between the variables. For understanding the relationship, one should look at correlation or regression analysis.