Random Sampling

1
Random Sampling

• Approximations of E(X), p.m.f,
• and p.d.f

2
Important

• Read through simulation slides for Thursday
• Homework 8 is due on Thursday
• Check web-page on Wednesday night -- print off
  any worksheets for simulation that might be there
  for Thursday
• Major mistakes on study guide put on-line there
  is a new one there now
• A different definition of the p.d.f and c.d.f for
  the uniform random variable then what was given
  in class but they mean the same thing.
• P.M.F versus P.D.F need clarification because I
  mispoke

3
P.M.F versus P.D.F

• Either graph can be a histogram
• I was assuming that the bin width will always be
  1 for a finite random variable but that is not
  necessarily the case
• Take X 0, ½, 1, etc.
• Probability Mass Function
• The values along the y-axis of a histogram
  represent probabilities
• If you sum up the probabilities, they should add
  up to 1 every time (regardless of the bin width)
• Thus, to determine is a graph is a p.m.f, you
  need to add up the heights of the rectangles if
  they add up to 1, then it is a p.m.f.

4
P.M.F versus P.D.F

• A probability density function can also be a
  histogram
• The values along the y-axis do not represent
  probabilities for a continuous random variable
• However, the area under graph must be equal to 1
• How can you check if a histogram represents an
  p.d.f? If the heights of the rectangles do not
  add up to 1, but the areas of the rectangles do
  sum to 1.

5
In conclusion

• Both a p.m.f and p.d.f graph can be histograms
• To tell if a histogram represents a p.m.f, the
  sum of the HEIGHTS of the rectangles must equal 1
  (because the heights represent probabilities)
• To tell if a histogram represents an p.d.f, the
  sum of the heights of the rectangles do not equal
  but the area of the rectangles do.
• Lets look at number 4b from the homework just
  turned in .

6
Why do we use Random Sampling?

• In business, we identify a random variable
• We want its probability information
• Problem We do not know its distribution OR
  expected value
• Solution Estimate E(X) and estimate FX(x) and
  fX(x) using random sampling

7
Definitions

• A number x that results from a trial of the
  process is called an observation of X
• A set x1, x2, , xn of n independent
  observations of the same random variable X is
  called a random sample of size n.

8
Example 1

• Suppose that X is the number of assembly line
  stoppages that occur during an 8-hour shift in a
  manufacturing plant. We could obtain a random
  sample of size 10 by watching the line for 10
  different shifts and recording the number of
  stoppages during each 8-hour shift

9
Example 1 (continued)

• Looking above, we see the information recorded
  during the 10 different shift observations
• We can compute the sample mean of the
  observations
• The sample mean is denoted by

10
Statistics and Probability

• There is a difference between probabilities and
  statistics even though people use them
  interchangably
• A number that describes a sample is called a
  statistic
• THEOREM The statistic can be used as an
  estimate of E(X).
• In general, the larger the sample size n, the
  better the estimate will be

11
Sample Mean

• We can find the mean of example 1

12
Approximating Probability Mass and Density
Functions

• If we have a large enough sample, we can
  approximate functions
• I.e., we can approximate a p.m.f or a p.d.f
  depending on the random variable
• If we approximate a p.m.f or p.d.f, we can also
  look at the corresponding graphs

13
Example 3

• Suppose that the assembly line discussed in
  Example 1 runs 24 hours a day, with workers in
  three shifts. Observations of the number of
  stoppages during an 8-hour shift were recorded
  for a nine month period. I.e., 819 different
  shifts were observed and recorded in the file
  Stoppages.xls.

14
Relative Frequencies

• Relative frequencies were plotted to obtain the
  histogram seen in Stoppages.xls
• The relative frequency of each value X in the
  sample gives an estimate for the probability that
  X will assume that value. WHY?
• How did we obtain the relative frequencies?
• A histogram will give a good approximation for
  the graph of fX

15
Continuous Random Variables

• A large random sample can also be used to
  approximate the p.d.f of a continuous random
  variable
• One way we can obtain our p.d.f is by looking at
  smaller and smaller bin widths of our data
• Use the HISTOGRAM function in Excel to find the
  approximation of the graph of the p.d.f

16
Example 4

• The manager of the plant that was described in
  the the previous examples wants to get a better
  of understanding of the delays caused by
  stoppagesof the assembly line. So, in addition
  to knowing how many stoppages there are, the
  manager wants to know how long they last.

17
Example 4 (continued)

• Let T be the (exact) length of time, in minutes,
  that a randomly selected stoppage will last
• QUESTION Is T a continuous random variable?
• In Stoppages.xls, the duration of each stoppage
  was also recorded for all 819 shifts
• Therefore, we have a random sample of
  observations for T

18
Example 4 (continued)

• Used the function HISTOGRAM in Excel to plot an
  approximation of the p.d.f., fT
• In Stoppages.xls, bin widths of 2 minutes are
  used
• Since our bin width is 2, to make the area under
  the graph be 1, we had to divide each relative
  frequency by 2 and then plot those new relative
  frequencies
• Note Here you are dividing the relative
  frequency by the bin width not the frequency by
  the bin width as stated in class
• Thus, you find the relative frequency as you did
  before and then divide it by the bin width
• By connecting the midpoints of the tops of the
  rectangles gives us an approximate curve

19
Using the approximated p.d.f

• We can use our plot to calculate probabilities
• For example, if we wanted to know P(2ltT?4), we
  could look at the corresponding area under the
  graph
• Note P(2ltT?4) corresponds to an area under the
  graph between (2,4 which is a rectangle
• So, to find our probability, find the area of the
  rectangle

20
Focus on the Project

• We have a continuous random variable Rnorm which
  gives the normalized ratio of weekly closing
  prices on Disney stock (class project)
• Option Focus.xls contains 417 values of Rnorm
  from 417 weekly closing ratios
• They are considered to be independent
  observations
• Thus, make up a random sample of size 417 for
  Rnorm

21
Focus on the Project

• We can calculate sample mean which we know should
  be equal to what?
• We can create a plot using the relative
  frequencies
• Note If your bin width is greater than 1, you
  will have to divide the relative frequency by
  your bin width to make the area under the curve
  be 1
• Graphing the midpoints at the tops of the bars
  will produce a line graph approximation for fnorm

22
What should you do?

• Plot an approximation of the probability density
  function for the normalized ratios of weekly
  closing prices
• The plot should be a line graph, where you are
  connecting the midpoints of the tops of the bars
• Remember, if your bin width is greater than 1 you
  will have to divide the relative frequencies by
  that width before you plot
• Find the sample mean of the normalized ratios
  you already know what it should equal