Probability problems day 14

  1. Let \(U_1, \dots, U_{100}\) be independent unif(0, 1) variables. What is the mgf of \(U_1 + U_2 + \cdots + U_{100}\)?

    1. What is the mgf of an exp(\(\lambda\)) distribution?
    2. Let \(X \sim \text{exp}(\lambda_1)\) and \(Y \sim \text{exp}(\lambda_2)\) be independent exponentials. What is the distribution of \(X + Y\)?

  3. Let \(U_1, U_2 \sim \text{unif}(0, 1)\) be independent uniform variables.

  1. What is the mgf of \(U_1 + U_2\)?
  2. Find the mgf of the variable with pdf \[ f(t) = \begin{cases} t & \text{for } 0 < t \leq 1 \\ 2 - t & \text{for } 1 < t < 2. \end{cases} \]
  3. What is the difference between the two mgfs that you found?
  1. Let \(Z\) be a standard normal variable. What is \(E[e^Z]\)?