Lecture 14: Thu, Oct 24

1
Lecture 14 Thu, Oct 24

• Announcements
• HW 5 due 2pm Friday, JMHH 414-1
• HW 6 due next Friday.
• Today Continuous Random Variables
• Probability density functions
• Probabilities, means, variances
• The Uniform Distribution

2
Continuous Distributions

• General continuous distributions
• Uniform distribution
• Normal distribution
• Exponential distribution
• Students T distribution
• Chi-squared distribution
• F distribution

3
Discrete vs. Continuous

• Discrete RV
• Takes a countable number of values
• Probability distribution set of point masses
• Calculate probabilities using summations
• Continuous RV
• Takes uncountable number of values
• Probability distribution is a density function
• Calculate probabilities using integrals

4
Continuous or Discrete RVs?

• The number of students that show up to Stat 101
  class today.
• The number of cheeseburgers consumed by Kevin
  Doyle during lunch break.
• The mileage driven on a tank of gasoline.
• The lifetime of a light bulb.
• The winning time at the Olympic 100 meter dash.

5
Graphs of Discrete and Continuous Distributions
Discrete Distribution
Continuous Distribution
6
From Histograms to Density Functions
7
Requirements for a Discrete Probability
Distribution

• 1) Probabilities must be between 0 and 1.
• 2) Probabilities must add up to 1

8
Requirements of a Probability Density Function

• 1) Density function f(x) is non-negative for all
  values of x
• 2) Total area under the curve f(x) is 1

9
The Density Function

• The density function f(x) is not a probability
  function itself.
• Rather, probabilities are given by the area under
  the density curve.

10
Calculating Probabilities

• For continuous RVs, the probability that X falls
  in a certain range is given by the area under the
  density curve

11
Probabilities

• For continuous RVs, the probability X takes on a
  single value is zero. That is,
• So, the equals sign can be ignored when computing
  probabilities

12
Shade in the Indicated Probabilities
13
Expected Value and Variancefor Continuous RVs

• The expected value is given by
• The variance is given by

14
General Formula

• For continuous RVs, the expected value of any
  function g(x) is given by

15
Shortcut Formula for Variance

• The expected value of
• The shortcut formula for variance is

16
Density/Graph Match-Up
Match up the following densities with their
graphs on the next page. Verify that these are
valid probability density functions.
17
(No Transcript)
18
Example

• Consider the function
• Draw a graph of the function
• Verify that this a valid probability density
  function.

19
Example

• The total daily demand for electricity, X, in
  Whitefish, Montana, has the following
  distribution (where X is measured in 100 KWHs)

20

• a) Plot the density function.
• b) Find the mean and variance of daily
  electricity demand in Whitefish.
• c) If the maximum daily electricity supply is
  100KWH, find the probability that demand will
  exceed supply.

21
(No Transcript)
22
The Uniform Density Function

• If X has a uniform distribution between the
  values a and b, then its density function is
  given by

23
Graph of the Uniform(1,3) Distribution
24
Quick Quiz

• a) What are the mean and median of this
  distribution?
• b) Find P(Xgt2).
• c) Find P(X1).
• d) Find P(2ltXlt3).

25
Mean and Variance of the Uniform Distribution

• If X is a uniform random variable on the interval
  a,b, the mean and variance are

26
Verification Using Integration
27
Example

• The bus arrives at your bus stop every 10
  minutes, but you dont know exactly when. Assume
  that the arrival time is a random variable with a
  uniform distribution.
• a) If you show up randomly at the bus stop, find
  the probability that the bus arrives
• i) Within 1 minute.
• ii) After 5 minutes.
• iii) Between 1 and 3 minutes.
• b) Find the expected waiting time and SD.

28
(No Transcript)
29
Example

• A hospital receives a pharmaceutical delivery
  each morning at a time that varies uniformly
  between 700 and 800am.
• a) What is the probability that the delivery on a
  given morning will occur between 715 and 730am?
• b) What is the expected time of delivery?
• c) Find the probability that the time of delivery
  will be within one standard deviation of the
  expected time that is, within the interval

30
(No Transcript)
31
Example

• The heights of Smurfs are known to be uniformly
  distributed between 20 and 30 inches.
• a) If a Smurf is randomly selected, what is the
  probability that she will be be taller than 28?
• b) If two Smurfs are randomly selected, what is
  the probability that at least one will be taller
  than 28?
• c) The Smurf-ball team only takes players in the
  tallest 25 of the population. What is the
  cutoff height for making the squad?
• d) What is the probability that a randomly
  selected player on the Smurf-ball team is taller
  than 29?

32
(No Transcript)