Probability Distributions: Part II

1
Probability Distributions Part II

• Continuous Random Variables
• Uniform Random Variables
• Exponential Random Variables
• Expected Value Revisited

2
Materials for Review and Practice

• Student Notebook
• Slides 73 thru 103 (pps. 147-162)
• Student Manual
• Pps. 140 - 167
• Continuous Random Variable Worksheet

3
Continuous Random Variables

• If the measurement scale of the rv X can be
  subdivided to any extent desired, then the
  variable is continuous if it cannot, the
  variable is discrete.
• For example
• if the variable is height or length, then it can
  be measured in kilometers, meters, centimeters,
  millimeters, and so on, so the variable is
  continuous.
• if X is the billing on a randomly selected
  monthly cell phone bill statement, then the
  smallest measurement is cents, so any value of X
  is a multiple of 0.01 and X is discrete.

4
Continuous Random Variables

• A random variable X is said to be continuous if
  its set of possible values is an entire interval
  of numbers.
• Some examples
• If in the study of the ecology of a lake, we make
  depth measurements at randomly chosen locations,
  then X the depth at such a location is a
  continuous random variable.
• If a chemical compound is randomly selected and
  its pH is determined, then X is a continuous rv
  because any pH between 0 and 14 is possible.

5
Probability Density Function (p.d.f)

• In the case of a discrete random variable, we
  referred to the probability distribution as a
  probability mass function or p.m.f., for short.
  If the random variable is continuous, we refer to
  the probability distribution as a probability
  density function or p.d.f. for short.
• The defining property of a p.d.f. is that the
  total area under the curve is equal to one.

6
Continuous Random Variables

• When a random variable is continuous, there are
  an infinite number of possible values for the
  r.v.
• Therefore, P(X a) 0 for any number a. This
  is true because if the P(X a) gt 0, and there
  are an infinite number of values for X, then the
  sum of the probabilities would be greater than
  one, which makes no sense.

7
p.m.f. versus p.d.f.
Source Thomson, R.B. and Lamoureux, C.G.
(2003). Mathematics for Business Decisions, Part
1 Probability and Simulation. MAA Washington,
DC
8
Comparison of c.d.f.s
Source Thomson, R.B. and Lamoureux, C.G.
(2003). Mathematics for Business Decisions, Part
1 Probability and Simulation. MAA Washington,
DC
9
Distributions of Continuous Random Variables

• Some Common Distributions
• Uniform Distributions
• Exponential Distributions
• Normal Distributions (Math 115b)
• Cauchy Distributions
• Chi-Squared Distributions
• Many Others

10
Uniform Probability Distribution

• Suppose that I take a bus to campus and that
  every five minutes a bus arrives at my stop.
  Because of variation in the time that I leave my
  house, I dont always arrive at the bus stop at
  the same time, so my waiting time X for the next
  bus is a continuous random variable. The set of
  possible values of X is the interval 0,5.

11
Uniform Probability Distribution

• If the waiting times are equally likely then the
  p.d.f. for X is

12
Uniform Probability Distribution

• A continuous random variable, X, is said to have
  a Uniform Distribution on the interval A,B if
  the p.d.f. of X is

13
c.d.f. for a Uniform Random Variable

• The cumulative distribution function F(x) for a
  uniform random variable, X, is defined for every
  number x by F(x) P(X ? x), that is, for each x,
  F(x) is the area under the density curve to the
  left of x.

14
Example of the graph of a p.d.f. and c.d.f. for a
Uniform Random Variable
Source Thomson, R.B. and Lamoureux, C.G.
(2003). Mathematics for Business Decisions, Part
1 Probability and Simulation. MAA Washington,
DC
15
Example One

• A security guard walks around your building, at
  random, from noon until 2 p.m., passing each
  office exactly one time. We assume that he is
  equally likely to pass your office at one time as
  at another during the two hours. What is the
  probability that he passes your office while you
  are away for lunch from 1215 p.m. until 112
  p.m.?

Source Thomson, R.B. and Lamoureux, C.G.
(2003). Mathematics for Business Decisions, Part
1 Probability and Simulation. MAA Washington,
DC
16
Example One (Continued)
Let T be the continuous random variable giving
the time from 1200 noon, when the guard passes
your office.
17
Example One (Continued)

• The guard is equally likely to pass your office
  at any given time. Thus, for any number a and
  any length of time t, such that 0 ? a?t ? 2, we
  have a?FT(t) a?P(T ? t) P(T ? a?t) FT(a?t).
  Over the interval 0 ? t ? 2, FT is a linear
  function.
• Wow!!! Do not be concerned if you do not
  understand these last two sentences out of the
  Student Notebook. It is an idea that goes beyond
  the contents of this course.

18
Example Two

• Upon studying low bids for shipping contracts, a
  microcomputer manufacturing company finds that
  intrastate contracts have low bids that are
  uniformly distributed between 20 and 25, in units
  of thousands of dollars.
• Find the probability that the low bid on the next
  intrastate shipping contract
• Is below 22,00
• Is in excess of 24,000
• Find the expected value of the low bids.

19
Exponential Distribution

• Exponential random variables usually describe the
  waiting time between consecutive events.
• In general, the p.d.f. and c.d.f. for an
  exponential random variable X

20
Exponential Distribution
21
p.d.f. and c.d.f. for an Exponential Random
Variable with alpha 2
22
Example Three

• Let Z be an exponential random variable with
  alpha 2. Compute
• fX(3)
• FX(3)
• P(X3)

23
Inverse c.d.f.

• For a exponential random variable, the c.d.f. is
  given by
• The inverse c.d.f. is

24
Expected Value Revisited

• What is the expected value of an exponential
  random variable?
• It can be shown that any exponential random
  variable X, with parameter ?, has E(X) ?.

25
Example Four

• Suppose that, on average, 30 customers per hour
  arrive at bank for service from the tellers. Let
  X be the continuous random variable that gives
  the time in minutes and parts of minutes,
  between the arrival of consecutive customers.
  Then X is an exponential random variable.
• Give a practical interpretation of FX (3)
• Compute P(X gt 2)

Source Thomson, R.B. and Lamoureux, C.G.
(2003). Mathematics for Business Decisions, Part
1 Probability and Simulation. MAA Washington,
DC
26
Expected Value

• Expected Value of a Discrete Random Variable
  is

27
Expected Value

• Let X be the binomial random variable from
  Example 2 on pp. 142, slide 63 of the Student
  Notebook. In this example, X gives the number of
  calls that are handled correctly, in a randomly
  selected set of 4 phone calls.

28
Expected Value (Binomial R.V.)

• In this example, the number of trials, n 4 and
  the probability of success is equal to 0.6.
• It can be shown that the expected value of a
  BINOMIAL RANDOM VARIABLE is given by
• E(X) n?p (4)(0.6) 2.40
• which is consistent with our previous result.

29
Expected Value

• We can use the probability density function to
  give us a geometric interpretation of the
  expected value (or mean) of a continuous random
  variable. The expected value is the point on the
  axis which perfectly balances the area to its
  right with the area to its left. This geometric
  interpretation will have to suffice until you
  learn a little calculus next semester! (This
  idea comes from physics and is known as the
  center of mass.)

30
Expected Value

• Given this geometric interpretation, what is the
  expected value (or mean) of a uniform random
  variable?