Spring 2018, Math 171

Week 8

  1. Exponential Distribution

    1. Let \(X \sim \mathrm{exp}(\lambda)\).

      1. Find the distribution of \(Y = \lceil X \rceil\)

        • (Answer) \(\mathrm{geometric}(1-e^{-\lambda})\)

      2. Show that \(X\) and \(Y\) are both memoryless

      3. Find the distribution of \(\beta X\)

        • (Answer) \(\mathrm{exponential}(\lambda/\beta)\)

      4. Find the distribution of \(e^{-X}\)

        • (Solution) Let \(Y = e^{-X}\). Note since \(X \in [0, \infty)\) we have \(Y \in (0, 1]\). \[\begin{aligned}F_Y(y) &= P(Y \le y)\\ &= P(e^{-X}\le y)\\&=P(-X \le \log(y))\\&= P(X \ge -\log(y))\\&=1-F_X(-\log(y))\end{aligned}\] \[\begin{aligned}f_Y(y) &= \frac{d}{dy}F_Y(y) \\ &= \frac{d}{dy}(1 - F_X(-\log(y))) \\ &= -f_X(-\log(y))\cdot \frac{-1}{y} \\ &= \frac{\lambda e^{-\lambda (-\log(y))}}{y} \\ &= \lambda y^{\lambda - 1}\end{aligned}\]

    2. Let \(U \sim \mathrm{uniform}[0,1]\). Find the distribution of \(-\alpha\log{U}\)

      • (Answer) \(\mathrm{exponential}(1/\alpha)\)

    3. Let \(X_1, X_2, \dots\overset{\mathrm{i.i.d}}{\sim} \mathrm{exp}(\lambda)\)

      1. (Discussed) Suppose \(N \sim \mathrm{geo}(p)\). Find the distribution of \(Z = \sum_{i=1}^N X_i\).

      2. Find the distribution of \(Q = \min(X_1, X_2, \dots X_n)\)

        • (Answer) \(\mathrm{exponential}(n\lambda)\)

      3. Find the cumulative distribution of \(V = \max(X_1, X_2, \dots X_n)\)

        • (Answer) \((1-e^{-v\lambda})^n\)

  2. Poisson Process Basics

    1. Let \(N(t)\) be a poisson process with rate \(\lambda\)

      1. (Discussed) Find the probability of no arrivals in \((3,5]\)

      2. (Discussed) Find the probability that there is exactly one arrival in each of the intervals: \((0,1], (1,2], (2,3], (3,4]\)

      3. (Discussed) Find the probability that there are two arrivals in \((0,2]\) and three arrivals in \((1,4]\)

      4. (Discussed) Find the covariance of \(N(t_1)\) and \(N(t_2)\) for \(0 < t_1 < t_2\)