Lecture 19

Qi Wang, Department of Statistics

Oct 3, 2018

- Bernoulli
- Binomial
- Hypergeometric
- Poisson
~~Geometric~~

- $X \sim Bern(p)$
**The definition of $X$**: the success of some event on a single trial.**Support:**$\{0, 1\}$**Parameter:**p**PMF:**$P_X(x) = p^x (1-p)^{1-x}$**Expected Value:**$E[X] = p$**Variance:**$Var(X) = p(1-p)$

- $X\sim Binomial(n, p)$
**The definition of $X$**: the total number of successes in a sequence of n independent Bernoulli experiments, with a success rate p**Support:**$\{0, 1, 2, \cdots, n\}$**Parameter:**n, p**PMF:**$P_X(x) = C_x^n p^x (1-p)^{n-x}$**Expected Value:**$E[X] = np$**Variance:**$Var(X) = np(1-p)$

- $X \sim Hyper(N, n, M)$
**The definition of $X$**: the number of success in $n$ trail without replacement from a finite population of size N that contains exactly M objects with that feature.**Support:**$\{0, 1, 2, \cdots, n\}$ or $\{0, 1, 2, \cdots, M\}$**Parameters:**- $N:$ Population size
- $M:$ Number of possible successes
- $n:$ Number of trials
**PMF:**$P_X(x) = \frac{C_x^M C_{n-x}^{N - M}}{C_n^N}$**Expected Value:**$E[X] = n \frac{M}{N}$**Variance:**$Var(X) = n \frac{M}{N} (1 - \frac{M}{N}) \frac{N-n}{N-1}$

- $X \sim Poisson(\lambda)$
**The definition of $X$**: the number of success per $\underline{\hspace{1cm}}$, and $\underline{\hspace{1cm}}$ can be time, length, space unit and so on**Support:**$\{0, 1, 2, \cdots\}$**Parameters:**$\lambda$, the average success rate per $\underline{\hspace{1cm}}$**PMF:**$P_X(x) = \frac{e^{-\lambda} \lambda^x}{x!}$**Expected Value:**$E[X] = \lambda$**Variance:**$Var(X) = \lambda$

In a recent year, the Wall Street Journal, reported that 58% of all American credit card holders had to pay a late fee. A random sample of 15 American credit card holders is selected. Let X be the number of credit card holders in the sample who had to pay a late fee. Assume that all credit card holders are independent of one another.

- State the distribution and parameter(s) for X. What is the support for X?
- What is the average and standard deviation of X?
- What is the probability that exactly 8 people in the sample had to pay a late fee?
- Given that at least one person in the sample had to pay a late fee, what is probability that 8 or 9 had to pay a late fee?

A college student is running late for his class and does not have time to pack his backpack carefully. He has 12 folders on his desk, 4 include HW assignments due today. He grabs 3 of the folders randomly and when he gets to class, counts the number of them that contain HW.

- What is the random variable here? What are the parameters?
- What is the expected number of folders with HW in them? What is the variance of X?
- What is the probability that 2 folders contain HW?
- What is the probability that fewer than 2 folders contain HW?

Courtney is running downtown and passes a gas station at a rate of once per 2.5 minutes (this is an average of 0.4 gas stations a minute). Let G be the number of gas stations Courtney passes during her thirty minute run.

- What is the support, distribution and parameter(s) of G?
- What is the probability Courtney passes 10 gas stations on her run?
- Courtney decides that to provide motivation, at the end of the run she will eat half a cookie for every gas station she passed on the run. What is the expected number of cookies Courtney will eat after the run?