1. Look up the distribution of a Poisson random variable with parameter $\lambda$, state it and use that to calculate $\mathbb{P}\left[X\ge 1\right]$, and $\mathbb{P}\left[X\ge 2\right]$ where $X$ is a Poisson random variable with parameter $1$.
2. Given that the m.g.f. ${\varphi }_X\left(t\right)$ of a Poisson random variable with parameter $\lambda$ is $e^{\lambda \left(e^t-1\right)}$, show that the sum of independent Poisson random variables $X_1$ with parameter ${\lambda }_1$ and $X_2$ with parameter ${\lambda }_2$ is again Poisson with parameter ${\lambda }_1+{\lambda }_2$.
3. Using the fact that the sum of two independent Poisson random variables with means ${\lambda }_1$ and ${\lambda }_2$ is again Poisson with mean ${\lambda }_1+{\lambda }_2$ find the exact probability that $\mathbb{P}\left[X_1+\cdots +X_{10}>15\right]$ where each $X_i$ is a Poisson random variable with parameter $1$.
4. Use the Markov Inequality to get a bound on $\mathbb{P}\left[X_1+\cdots +X_{10}>15\right]$ where each $X_i$ is a Poisson random variable with parameter $1$.

where $X_1,X_2,\dots$ are independent, identically distributed continuous random variables with mean $0$ and variance ${\sigma }^2$. (Note that this is an additive assumption about the change in a stock price. In the binomial tree models, we assumed that a stock’s price changes by a multiplicative factor up or down. We will have more to say about these two distinct models later.) Suppose that a stock’s price today is $100$. If ${\sigma }^2=1$, what can you say about the probability that after $10$ days, the stock’s price will be between $95$ and $105$ on the tenth day?