# 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 $$0 < t_1 < t_2$$