```help probcalc
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Title:

probcalc: Probability Calculator for Binomial, Poisson, and Normal Distributio
> ns

Syntax

probcalc {dist param1 param2 param3}, param4 param5

Description:

probcalc calculates the probability mass function for the discrete binomial
and Poisson distributions and the probability density function for the
continuous normal distribution.  Output is written to the display in a
format useful for learning probability calculations.

The algorithm can handle probability questions pertaining to, for example,
"exactly 5 events," "at most 120 events," and "at least 7 events."

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Commands for the Binomial Distribution:

Probability of observing exactly x events, P(X=x)

probcalc b #n #p exactly #x} (Note: bold represents commands and
#param represents actual numeric input value)

The probability distribution based on n and p, densities only shown for
pmf>0.01

probcalc b #n #p dist

Probability of observing at most x events, P(X<=x)

probcalc b #n #p atmost #x

Probability of observing at least x events, P(X>=x)

probcalc b #n #p atleast #x

Commands for the Poisson Distribution:

Probability of observing exactly x events, P(X=x)

probcalc p #mu exactly #x

The probability distribution based on mu, densities only shown for
pmf>0.01

probcalc p #mu dist

Probability of observing at most x events, P(X<=x)

probcalc p #mu atmost #x

Probability of observing at least x events, P(X>=x)

probcalc p #mu atleast #x

Commands for the Normal Distribution:

Probability of observing a value of X between a and b, P(a<=X<b)

probcalc n #mean #sigma between #a #b

The probability density within plus-minus 4 standard deviations of a mean
(30 bins)

probcalc n #mean #sigma dist

Probability of observing an X-value that is at most x, P(X<=x)

probcalc n #mean #sigma atmost #x

Probability of observing an X-value that is at least x, P(X>=x)

probcalc n #mean #sigma atleast #x

Examples:

A set of measurements for a particular variable follow the binomial
distribution with parameters n=15 and p=0.15. What is the probability of
occurrence of exactly 5 events, P(X=5)?

.probcalc b 15 0.15 exactly 5

Generate the distribution of binomial variates based on the parameter
values n=30 and p=0.35. What is the probability distribution, showing
only variates for which pmf>0.01?

.probcalc b 30 0.35 dist

A variable was observed to follow the binomial distribution with n=100
and p=0.17. What is the probability of observing at most 16 events,
P(X<=16)?  (16 and less --> left tail)

.probcalc b 100 0.17 atmost 16

Quark-gluon collisions were tabulated for a number of identical collider
experiments and were found to follow a binomial distribution with n=1500
and p=0.25. What is the chance that at least 375 collisions would have
been observed, P(X>=375)?  (375 and greater --> right tail)

.probcalc b 1500 0.25 atleast 375

An event occurs at a rate of mu=15 times per day on average. What is the
probability of exactly 5 events occuring on any given day, P(X=5)?

.probcalc p 15 exactly 5

What is the Poisson probability distribution when mu=10? (Note: only
variates for which pmf>0.01 are shown)

.probcalc p 10 dist

An event occurs on average mu=100 per millisecond. What is the
probability of at most 93 events will be observed in a millisecond,
P(X<=93)?  (93 and less -> left tail)

.probcalc p 100 atmost 93

The number of ions that interact within a square centimeter of target
material is mu=14.  What is the probability that at least 12 ions will
interact in a square centimeter of area, P(X>=12)?  (12 and greater -->
right tail)

.probcalc p 14 atleast 12

Daily caloric intake among a set of low-fat diet participants was found
to be normally distributed with mean 1987 (calories) and s.d.=52. What
proportion of participants would be expected to have daily caloric intake
values between 1930 and 2040 calories, P(1930<X<=2040)?

.probcalc n 1987 52 between 1930 2040

Patient weight measurements indicate a mean of 150 and s.d. of 20.  What
is the normal probability density between plus-minus 4 standard
deviations of the mean?

.probcalc n 150 20 dist

Weight of high school students was determined to be normally distributed
with mean=128 (lbs) and s.d.=25 (i.e., s.d.).  What proportion of
students are likely to weigh less than 120 lbs, P(X<=120)?

.probcalc n 128 25 atmost 120

Daily temperature in Houston was determined to be normally distributed
with mean=68 and s.d.=16.  What is the chance of the temperature being
90F or greater, P(X>=90)?

.probcalc n 68 16 atleast 90

During run-time, output results merely displayed to the screen, for
cutting and pasting.

Author:

Leif E. Peterson
Associate Professor of Public Health
Weill Cornell Medical College, Cornell University
Center for Biostatistics, The Methodist Hospital Research Institute (TMHRI)
Email: lepeterson@tmhs.org

Also see

Online: binomialp, binomial, poissonp, poisson, normal.

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