Random Sampling: Estimating Probabilities and Expected Values - Prof. L. Dawson, Study notes of Mathematics

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Random SamplingRandom SamplingRandom SamplingRandom Sampling

Math 115ASpring 2008Dawson

Why random samples?Why random samples?^
^ We would like to have probability informationfor our random variable

X
^ Often we don’t know the distribution of

X^ or even its expected value
To estimate the expected value,

For^ fweX^ X^
^ To estimate the expected value,

For^ fweX^ X^ use^ random sampling

. ◦^ We may approximate the distribution if we can takea large enough sample of

n^ independent observations of^ X ◦ We call these^ n^ independent observations of

X, {x, x, x, …, x}, a random sample of size n^123 n

.

ApproximatingApproximating

p.m.f.p.m.f.ss
^ To approximate a probability mass function:^ ◦^ Group the data according to values of

X ◦^ Create a histogram for these groupings ◦^ Calculate the relative frequencies

Sample Data 0.2000.1800.1600.1400.1200.1000.0800.060 Relative Frequency 0.0400.020 0.000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14^ Stoppages

Discrete Random VariableDiscrete Random Variable^
^ Example:^ Let^ X

be the denomination of a chipselected at random from a box that has 4 $1chips, 3 $5 chips, 2 $25 chips, and 1 $100chip. There are twenty observations of

X given below. Use the sample to plot an approximation of the

p.m.f.^ of^ X^ and to approximation of the

p.m.f.^ of^ X^ and to estimate^ E(X).^ (note: changed data from in class to match next slide)^ $25^ $^

$25^ $1^ $5 $25 $1 $1^ $1^ $5 $5 $1 $25^ $1^ $100$1 $1 $25^ $5^ $

ApproximatingApproximating

p.d.f.p.d.f.ss
^ To approximate a probability densityfunction:^ ◦^ Group the data according to the range ofvalues of^ X^ ◦^ Create a histogram for these groupings◦^ Create a histogram for these groupings^ ◦^ Calculate the relative frequencies^ ◦^ Divide the relative frequencies by the bin widthto calculate the

p.d.f.^ heights ◦ Create a line plot connecting the midpoints ofeach of the columns

ApproximatingApproximating

p.d.f.p.d.f.ss^ 0. Rel.Freq^ 0.

25. 0 125.^0 = 2 2125.^0 ×=× wl^25.^0 =

(bins are labeled by their midpoint)^ 0. (^1 3 5 7 9 11 13) In a plot of relative frequencies,the probability is given by theheight of the rectangle. But thatshould not be true for a^ p.d.f.^ – itmust be the area that is equal tothe probability.

(^1 3 5 7 9 11 13) By dividing the relativefrequency by the bin width, weare ensuring that the area ofeach rectangle (length timeswidth) is equal to the relativefrequency.

Continuous Random VariablesContinuous Random Variables^ Example:^ A bus arrives every 10 minutes.

Let^ W be the waiting time (in minutes) until the nextbus.^ Fifty observations of

W^ are given below. Use the sample to plot an approximation of the p.d.f.^ of^ W^ and to estimate

E(X). p.d.f.^ of^ W^ and to estimate

E(X). 9.7^ 0.8^ 6.5^ 5.5^ 9.

3.4^ 7.5^ 4.8^ 0.2^ 8.

7.0^ 2.0^ 0.6^ 2.0^ 3.

2.4^ 8.0^ 6.2^ 6.1^ 5.

0.3^ 4.5^ 9.4^ 9.2^ 3.

0.9^ 5.9^ 7.8^ 5.6^ 1.

2.3^ 1.3^ 9.8^ 7.2^ 6.

6.6^ 5.1^ 3.7^ 0.5^ 5.

8.2^ 3.5^ 3.3^ 3.1^ 7.

3.8^ 6.4^ 4.4^ 8.7^ 9.

Continuous Random VariablesContinuous Random Variables^ Example (cont’d.):^ Bin^ Frequency

BinRelativeMidpointFrequency^ (frequencydivided by^ total)

Height^ (relativefrequencydivided by binwidth) total) 2 9

1 0.^

3 0.^

5 0.^

7 0.^

9 0.^

Sum^50

1.00^.

BootstrappingBootstrapping^
^ Def: Bootstrapping

is sampling with replacement fromone sample to generate new samples, typically forthe purpose of estimating probabilities andparameters.
^ Why?
^ Why?^ ◦^ We are not very likely to have a large enoughsample set, due to costs and time.^ ◦^ We can simulate a larger data set by sampling fromour original data, this is bootstrapping

.
^ We will use the

Excel^ functions, VLOOKUPand RANDBETWEEN to help bootstrapping.

VLOOKUPVLOOKUP^
^ VLOOKUP is found under the “Formulas” Tab, “Lookup &Reference”

RANDBETWEENRANDBETWEEN^
^ RANDBETWEEN returns a random integer between the twonumbers you input^
^ RANDBETWEEN is found under the “Formulas” Tab, “Math &Trig”

RANDBETWEENRANDBETWEEN

^ Bottom:^ thelower integer inyour intervalwhich you are^ searchingsearching
^ Top:^ the upperinteger in yourinterval whichyou are searching

ExampleExample^
^ The times at which 270 calls arrived at a company’sswitchboard are shown in the sheet

Log^ of^ Phone Log.xls.
Let^ T^ be the random variable that gives the time (inminutes) until the arrival of the first call andbetween the arrivals of successive calls.
The 270 times in the sheet

Log^ determine 270 time intervals, which may be assumed to represent independent^ observations of

T.
^ Let^ L^ be the random variable that gives the arrivaltime of the last call in a run of 15 calls, starting at9 a.m.

Example (cont’d.)Example (cont’d.)^
^ Use^ VLOOKUP

and^ RANDBETWEEN

to generate 20 observations of

L.
^ In the sheet^ Random

of^ My Phone Log.xls

, RANDBETWEEN^

is used to randomly select an integer between 1 and 270.
In the sheet^ Times

of^ My Phone Log.xls

, VLOOKUP^ is used to find the correspondingtime between calls.
In the sheet^ Start Times

, the time between calls is added to the beginning of the hour (orthe arrival time of the previous call).