# Sum of 2 Non-identical Uniforms RVs?

1. Aug 24, 2010

### zli034

Does anyone know to formulate the pmf and pdf of sum 2 uniform random variables of non-identically distributed?

Say rv X is uniformly distributed range (0,1), and rv Y is uniformly distribute range (-9,0).
For Z = X+Y, what is the probability distribution of Z?

Thanks in advance. So many things are just nice to know.

2. Aug 24, 2010

### Mute

For independently distributed random variables $x_i$, the distribution of a sum of these variables,

$$z = a_1x_1 + \dots a_N x_N$$

can be found using characteristic functions. The characteristic function of a random variable is given by

$$\varphi_{X_k}(t) = \langle \exp(ix_k t)\rangle$$
i.e., the expectation value of exp(ix_k t). Note that this is just a fourier transform for continuous variables and a fourier series for discrete variables, so the original pdf can be recovered by performing an inverse fourier transform.

Given this, the characteristic funtion of z is

$$\varphi_Z(t) = \prod_{k=1}^N \varphi_{X_k}(a_kt)$$

So, to get the pdf of z, inverse fourier transform the resulting expression.

Try applying this to your specific case of two different uniform variables. (I'm not sure off the top of my head if the inverse FT can be performed in closed form. It will involve the product of two sinc functions. Intuitively, though, I would expect the result for the specific example to be the two separate uniform distributions stitched together and the area renormalized to 1, since the ranges do not overlap)