Compute the leads in coin tossing probability mass function.
The formula for the leads in coin tossing probability mass
with n a non-negative integer denoting the shape
Given 2n coin tosses, let 2k denote the last
coin toss for which the cumulative number of heads and
tails were equal (0 ≤ k ≤ n). The
leads in coin tossing distribution is the distribution of
k. We use 2n and 2k since a tie can
only occur on an even number toss.
Feller (see Further Information below) points out that the
common belief that in a long coin-tossing game that each
player will be on the winning side approximately half the
time with frequent lead changes is in fact erroneous.
Frequent lead changes implies that k should be
close to n. However, for this distribution the
probability that x = k is equal to the
probability that x = n - k. The
implication of this is that with probability 1/2 no
equalization will occur in the second half of the game
regardless of the length of the game.
The mean and variance of the leads in coin tossing
The leads in coin tossing distribution is also known as the
discrete arcsine distribution.
LET <y> = LCTPDF(<x>,<n>)
where <x> is a variable, a number, or a parameter containing
values between 0 and <n>;
<n> is a number or parameter that defines the upper
limit of the leads in coin tossing distribution;
<y> is a variable or a parameter (depending on what
<x> is) where the computed pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = LCTPDF(3,20)
LET Y = LCTPDF(X,100)
For a number of commands utilizing the leads in coin tossing
distribution, it is convenient to bin the data. There
are two basic ways of binning the data.
- For some commands (histograms, maximum likelihood
estimation), bins with equal size widths are required.
This can be accomplished with the following commands:
LET AMIN = MINIMUM Y
LET AMAX = MAXIMUM Y
LET AMIN2 = AMIN - 0.5
LET AMAX2 = AMAX + 0.5
CLASS MINIMUM AMIN2
CLASS MAXIMUM AMAX2
CLASS WIDTH 1
LET Y2 X2 = BINNED
- For some commands, unequal width bins may be
helpful. In particular, for the chi-square goodness
of fit, it is typically recommended that the minimum
class frequency be at least 5. In this case, it may
be helpful to combine small frequencies in the tails.
Unequal class width bins can be created with the
LET MINSIZE = <value>
LET Y3 XLOW XHIGH = INTEGER FREQUENCY TABLE Y
If you already have equal width bins data, you can
use the commands
LET MINSIZE = <value>
LET Y3 XLOW XHIGH = COMBINE FREQUENCY TABLE Y2 X2
The MINSIZE parameter defines the minimum class
frequency. The default value is 5.
You can generate leads in coin tossing random numbers,
probability plots, and chi-square goodness of fit tests
with the following commands:
LET NSIZE = VALUE
LET N = <value>
LET Y = LEADS IN COIN TOSSING RANDOM NUMBERS ...
FOR I = 1 1 N
LEADS IN COIN TOSSING PROBABILITY PLOT Y
LEADS IN COIN TOSSING PROBABILITY PLOT Y2 X2
LEADS IN COIN TOSSING PROBABILITY PLOT Y3 XLOW XHIGH
LEADS IN COIN TOSSING CHI-SQUARE GOODNESS OF FIT Y
LEADS IN COIN TOSSING CHI-SQUARE GOODNESS OF FIT Y2 X2
LEADS IN COIN TOSSING CHI-SQUARE GOODNESS OF FIT ...
Y3 XLOW XHIGH
Dataplot does not provide any explicit parameter estimation
methods. For this distribution, we simply subtract the
minimum value (so the data starts at zero) and then use
the maximum value as the estimate of N. We can then apply
goodness of fit tests (i.e., the probability plot or the
chi-square goodness of fit) to see if the leads in coin tossing
is an appropriate distribution.
= Compute the leads in coin tossing cumulative
= Compute the leads in coin tossing percent point
= Compute the discrete uniform probability mass function.
= Compute the matching probability mass function.
= Compute the lost games probability mass function.
= Compute the arcsine probability density function.
= Compute the beta probability density function.
= Compute the uniform probability mass function.
Feller (1957), "Introduction to Probability Theory",
Third Edition, John Wiley and Sons, pp. 78-84.
Johnson, Kotz, and Kemp (1992), "Univariate Discrete
Distributions", Second Edition, Wiley, pp. 274-275.
TITLE CASE ASIS
TITLE Leads in Coin Tossing Probability Mass Function CR() ...
(N = 50)
LABEL CASE ASIS
Y1LABEL Probability Mass
TIC OFFSET UNITS SCREEN
TIC OFFSET 3 3
PLOT LCTPDF(X,50) FOR X = 0 1 50
Date created: 6/20/2006
Last updated: 6/20/2006
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