Chapter 4 Review

Exercise 4.1 Elena had 63 popsicle sticks. She organized them into 6 bundles of the same size, and had 3 loose ones left. Boyan gave her 2 more bundles of popsicle sticks of the same size that she already had, plus 8 loose ones. How many popsicle sticks does Elena have now?

Exercise 4.2 Let $u(t)$ denote the temperature of an object at time $t$. Let $A$ denote the ambient temperature around the object, assumed constant. Consider the following physics law: The rate at which the object temperature changes is proportional to the difference between the object temperature and the ambient temperature. Translate the plain English below into math:

1. What is a simple mathematical expression for the rate at which the object temperature changes?
2. What is a simple mathematical expression for the difference between the object temperature and the ambient temperature?
3. Write an equation to express that the two quantities in (1) and (2) are proportional to each other. Use $k$ for your constant of proportionality.
4. What should the sign of the constant $k$ be to make sense, given the nature of cooling? Explain your reasoning.

Exercise 4.3 Let $X$ be a Bernoulli random variable with possible values 0 and 1, where 1 corresponds to success and 0 to failure. Assume that the probability of success is $p$. Find $E[X]$, the expected value of $X$.

Exercise 4.4 For the same Bernoulli random variable, as given in problem 1, find $Var(X)$, the variance of $X$.

Exercise 4.5 Roll a fair die and define success as rolling a 6. Consider the Bernoulli random variable with success so defined. What are the probabilities of success and failure for this Bernoulli random variable?

Exercise 4.6 Let $N$ be a Poisson random variable with rate $\lambda$. Use the Poisson probability distribution, given in the previous lecture, to verify that all probabilities sum to 1. In other words, verify that it holds

$$\sum _{k=0}^{\mathrm{\infty }}P(N=k)=1$$

where:

$$P(N=k)=e^{-\lambda }\frac{{\lambda }^k}{k!}$$