Math 6010 • Linear Models

Let \(\alpha,\lambda\) be fixed, strictly positive, and finite. The
Gamma\((\alpha,\lambda)\) density function is
\[
  f(x):=
  \begin{cases}
    \dfrac{\lambda^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x},
      & \text{if } x>0,\\
    0, & \text{if } x\leq0,
  \end{cases}
\]
where \(\Gamma(\alpha):=\int_0^\infty t^{\alpha-1}e^{-t}\,dt\). This defines a
probability density function.

\textbf{(a)} Compute the characteristic function of \(f\).

\textbf{(b)} Suppose \(X_1,\ldots,X_n\) are independent with common density
Gamma\((1,\lambda)\). Prove that \(S_n:=X_1+\cdots+X_n\) has a Gamma density; also
compute its parameters.

\textbf{(c)} Compute \(\Prob\{S_n<k<S_{n+1}\}\) for all integers \(k\geq1\).