Lecture 25

Qi Wang, Department of Statistics

Oct 22, 2018

At a high school track and field tournament, Mark’s high jumps vary evenly from 1.8 meters to 2.15 meters, while Dan’s high jumps vary evenly from 1.75 to 2.3 meters.

- Let M be the length of one of Mark’s high jumps. What are the distribution and parameter(s) of M?
- What is that the probability that Mark jumps between 1.88 and 2.05 meters?
- Which jumper’s jumps has the smaller standard deviation?
- What is the probability that one of Dan’s high jumps is exactly 2.0 meters?
- What length cuts off the highest 25% of Dan’s high jumps?

**The definition of $X$**: The waiting time until the first success**Support:**$X \in [0, +\infty)$ or $X \ge 0$**Parameter:**$\mu$, the average amount of time for one success**PDF:**$f_X(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}}$, for $ x \ge 0$**CDF:**\[ F_X(x)= P(X \le x) = \left\{ \begin{array}{ll} 0, x < 0\\ 1 - e^{-\frac{x}{\mu}}, x \ge 0 \end{array} \right. \]**Expected Value:**$E[X] = \mu$**Variance:**$Var(X) = \mu^2$**Notation:**$X \sim Exp(\mu)$ or $X \sim Exponential(\mu)$

If $X \sim Exp(\mu)$

- Tail Probability formula: $P(X > x) = e^{-\frac{x}{\mu}}$
- Memoryless Property: $P(X > s + t| X > s) = P(X > t)$

It is your birthday and you are waiting for someone to write you a birthday message on Facebook. On average (on your birthday) you receive a facebook message every 10 minutes. Assume that birthday messages arrive independently.

- What is the probability the next posting takes 15 minutes or longer to appear? What distribution, parameter(s) and support are you using?
- What is the standard deviation of the time between birthday postings?
- What is the probability that it takes 12.5 minutes for the next birthday posting?
- Suppose that the most recent birthday posting was done at 1:40 pm and it is now 1:45 pm. What is the probability that you will have to wait until 1:53 pm or later for the next message?
- What is the probability that your wait time for the next three messages is less than 8 minutes?
- What is your median waiting time for birthday messages?

**The definition of $X$**: The waiting time until the first success**Support:**$X \in [0, +\infty)$ or $X \ge 0$**Parameter:**$\lambda$, the number of success per time unit, $\lambda = \frac{1}{\mu}$**PDF:**$f_X(x) = \lambda e^{- \lambda x}$, for $ x \ge 0$**CDF:**\[ F_X(x)= P(X \le x) = \left\{ \begin{array}{ll} 0, x < 0\\ 1 - e^{-\lambda x}, x \ge 0 \end{array} \right. \]**Expected Value:**$E[X] = \frac{1}{\lambda}$**Variance:**$Var(X) = \frac{1}{\lambda^2}$**Notation:**$X \sim Exp(\lambda)$ or $X \sim Exponential(\lambda)$

You are teaching your new puppy to fetch a ball and are interested in the amount of time it takes for the puppy to run, get the ball and bring it back to you after you throw it.

- Scenario 1: The average amount of time it takes is 30 seconds.
- Scenario 2: The amount of time it takes is anywhere from 10 to 45 seconds.

- What is the distribution, parameter(s) and support for Scenario 1 and Scenario 2
- For each scenario, find the probability that it takes more than 22 seconds for the puppy to fetch the ball. Less than 50 seconds??
- What is the probability that it will take the puppy less than 40 seconds to fetch the ball knowing that it took the puppy longer than 22 seconds
- Assuming independence, what is the probability that it takes the puppy less than 40 seconds to fetch each of the next 5 balls?