$\begingroup$Given independent and identically distributed random variables $X_1, X_2, \dots, X_n$, each of them has the same p.d.f $f(x) = Pr(X = x)$ on support $(a, b)$.
How do I find the pdf or cdf of $Y = \sum_{i = 1}^n a_iX_i$, where $1 \le i \le n$ and $a_i$'s are constants?
$\endgroup$1 Answer
$\begingroup$The pdf of $\alpha_iX_i$ is $\frac{1}{|\alpha_i|}f(\frac{x}{\alpha_i})$ which we shall denote $g_i(x)$. Then since the $\alpha_iX_i$ are independent, the pdf of the sum Y is the convolution of the $g_i$:
$$f_Y(y) = (g_1 * g_2 * g_3 * \dots *g_n)(y)$$
where * means convolution.
$$(g_i * g_j)(y) = \int_{-\infty}^{\infty}g_i(t)g_j(y-t)dt$$
for continuous random variables, while for discrete random variables it would be
$$(g_i * g_j)[n] = \sum_{k = -\infty}^{\infty}g_i[k]g_j[n-k].$$
$\endgroup$2
