Introduction to Probability Models

Lecture 29

Qi Wang, Department of Statistics

Nov 1, 2017

Reminder

  1. Exam 2 will be from 8:00pm to 9:30pm at CL50 on Thursday, November 2nd
  2. There still will be class this Friday

Named Random Variables

  • Discrete
    • Bernoulli
    • Binomial
    • Hypergeometric
    • Poisson
    • Geometric
    • Negative Binomial
  • Continuous
    • Uniform
    • Exponential
    • Normal
  • Refer to Lecture 21 for a summary of discrete random variables

Uniform Random Variable

  • The definition of $X$ : the variable is evenly distributed over an interval
  • Support: $X \in [a, b]$ or $a \le X \le b$
  • Parameter: $a, b$, the end points of the interval
  • PMF: $f_X(x) = \frac{1}{b- a}$, for $a \le x \le b$
  • CDF: \[ F_X(x)=\left\{ \begin{array}{ll} 0, x < a\\ \frac{x - a}{b - a}, a \le x \le b\\ 1, x > b \end{array} \right. \]
  • Expected Value: $E[X] = \frac{a + b}{2}$
  • Variance: $Var(X) = \frac{(b - a)^2}{12}$
  • Notation: $X \sim Unif(a, b)$

Exponential Random Variable

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

Example 1

Suppose that a continuous random variable, X, has the probability density function (PDF) given below: \[ f_X(x)=\left\{ \begin{array}{ll} \frac{3}{2}x, 0 \le x \le 1\\ \frac{1}{4}, 5 \le x \le 6\\ 0, otherwise \end{array} \right. \]

  1. What is the probability that $X$ is equal to 4?
  2. What is the probability that $X$ is more than 4?
  3. Find $F_X(5.6)$
  4. Knowing that $X$ is more than $0.8$, what is the probability that $X$ is less than 5.6?
  5. What is the $85_{th}$ percentile of $X$?

Example 2

Professor Pine is a Pokémon professor who specializes in Pokémon health and has a clinic to treat sick/fainted Pokémon . The clinic is open 12 hours each day (9:00 am to 9:00 pm). On average, 15 Pokémon arrive and are treated at the clinic during its open hours.

  1. Let H be the number of Pokémon treated in 1 hour. What is the probability that Professor Pine treats at least two Pokémon in the next hour? What are the support, distribution, and parameter(s) of H?
  2. Professor Pine’s graduate assistant, Joy, has been keeping track of the amount of time between Pokémon arrivals at the clinic. What is the probability that the time between two consecutive arrivals is under 90 minutes? What distribution and parameter(s) are you using?
  3. Seven Pokémon arrived at the clinic between 3:00 pm and 8:00 pm. What is the probability that 3 of those Pokémon arrived between 5:00 and 7:00 pm?

Example 3

A certain component that your company manufactures has a weight that varies evenly from 59 grams to 75 grams. Each of these manufactured components is independent of the others.

  1. What is the probability that a randomly chosen component weighs between 62 and 73 grams? State the distribution and parameter(s) you are using.
  2. What is the standard deviation of the component weights?
  3. Find the cut-off for the upper quartile of component weights.
  4. Knowing that a randomly chosen component weighs less than 70 grams, what is the probability that it weighs more than 64.8 grams?

Example 4

Suppose that you have a box containing 50 balls, 35 of which are black. The rest are red.

  1. You will randomly choose 10 balls without replacement. Let R be the number of red balls in your sample. What is the probability that there are at least 2 red balls in your sample? What are the distribution and parameter(s) for R? .
  2. After the sampling in part a) you will put all the balls back in the box. You will now randomly choose a ball and note its color. Then you will return the ball to the box. (i.e. You are sampling with replacement.) Let T be number of balls you draw out until you get your first red ball. What is the average number of tries that it will take to get your first red ball? What are the distribution and parameter(s) of T?
  3. You have drawn out 3 balls (with replacement) and have still not gotten a red ball. What is the probability that it will take at most 5 draws to get your first red?
  4. What is the probability that it takes 9 tries (with replacement) to get three red balls?