What is a good way to resolve this?

Question

The random variable X has a probability density function fX (x) = 1/3 δ(x)+ 2/3 δ(x−2). Another random variable, Y , has a probability density function fY (y) = 1/2 δ(y) + 1/2 δ(y − 3). X and Y are statistically independent. Obtain an analytical expression for the probability density function of Z, where Z = X + Y . Also, sketch and completely label your result.

Answer 1

The probability distribution of the sum of two independent variables with probability distributions requires a convolution integral:

f_z(z) = Integral from -inf to inf of (f_x(z-y)*f_y(y)dy) or the same integral with x and y switched.

There will be four delta functions that have to be multiplied together and integrated. The first term after distributing will be

integral of 1/3 δ(z-y) * 1/2 * δ(y)dy = 1/6 δ(z) Multiply coefficients and δ(y) only exists at y = 0, so plug y = 0 into other term in order to get z function.

Good luck!