Maintenance script: Automated superscript correction

2016-02-24T22:25:47Z

Automated superscript correction

← Older revision		Revision as of 22:25, 24 February 2016
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	# cumSum(binomcdf(N,0 gives the list {1 2 ... N+1}, and cumSum(not(binompdf(N,0 gives the list {0 1 2 ... N}.		# cumSum(binomcdf(N,0 gives the list {1 2 ... N+1}, and cumSum(not(binompdf(N,0 gives the list {0 1 2 ... N}.
	# With seq(, you normally do math inside the list: seq(3I^^2^^,I,0,5		# With seq(, you normally do math inside the list: seq(3I<sup>2</sup>,I,0,5
	# With these commands, you do the same math outside the list: 3Ans^^2^^ where Ans is the list {0 1 ... 5}.		# With these commands, you do the same math outside the list: 3Ans<sup>2</sup> where Ans is the list {0 1 ... 5}.

KermMartian at 19:25, 24 February 2016

2016-02-24T19:25:59Z

← Older revision		Revision as of 19:25, 24 February 2016
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	Since "at most N" is equivalent to "0 or 1 or 2 or 3 or ... N", and since we can combine these probabilities by adding them, we can come up with an expression for binomcdf( by adding up values of [[TI-BASIC:Binompdf\|Binompdf(]]:		Since "at most N" is equivalent to "0 or 1 or 2 or 3 or ... N", and since we can combine these probabilities by adding them, we can come up with an expression for binomcdf( by adding up values of [[TI-BASIC:Binompdf\|Binompdf(]]:

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{binomcdf}(n,p,k) = \sum_{i=0}^{k}\operatorname{binompdf}(n,p,i) = \sum_{i=0}^{k}\binom{n}{i}\,p^i\,(1-p)^{n-i}		\operatorname{binomcdf}(n,p,k) = \sum_{i=0}^{k}\operatorname{binompdf}(n,p,i) = \sum_{i=0}^{k}\binom{n}{i}\,p^i\,(1-p)^{n-i}
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	(If you're not familiar with sigma notation, <math>\sum_{i=0}^{k}</math> just means "add the following up for each value of i 0 through k")		(If you're not familiar with sigma notation, <math>\sum_{i=0}^{k}</math> just means "add the following up for each value of i 0 through k")

Maintenance script: Initial automated import

2016-02-24T18:38:30Z

Initial automated import

New page

{{Template:TI-BASIC:Command
|picture=BINOMCDF.GIF
|summary=Calculates the binomial cumulative probability, either at a single value or for all values
|syntax=for a single value:
binomcdf(''trials'', ''probability'', ''value''

for a list of all values (0 to ''trials'')
binomcdf(''trials'', ''probability''
|location=Press:
# 2ND DISTR to access the distribution menu
# ALPHA A to select binomcdf(, or use arrows.
Press ALPHA B instead of ALPHA A on a TI-84+/SE with OS 2.30 or higher.
|compatibility=TI-83/84/+/SE
|size=2 bytes
}}

This command is used to calculate the binomial cumulative probability function. In plainer language, it solves a specific type of often-encountered probability problem, that occurs under the following conditions:
# A specific event has only two outcomes, which we will call "success" and "failure"
# This event is going to repeat a specific number of times, or "trials"
# Success or failure is determined randomly with the same probability of success each time the event occurs
# We're interested in the probability that there are '''at most''' N successes

For example, consider a couple that intends to have 4 children. What is the probability that at most 2 are girls?

# The event here is a child being born. It has two outcomes "boy" or "girl". In this case, since the question is about girls, it's easier to call "girl" a success.
# The event is going to repeat 4 times, so we have 4 trials
# The probability of a girl being born is 50% or 1/2 each time
# We're interested in the probability that there are at most 2 successes (2 girls)

The syntax here is binomcdf(''trials'', ''probability'', ''value''). In this case:

:binomcdf(4,.5,2

This will give .6875 when you run it, so there's a .6875 probability out of 4 children, at most 2 will be girls.

An alternate syntax for binomcdf( leaves off the last argument, ''value''. This tells the calculator to compute a list of the results for all values. For example:

:binomcdf(4,.5

This will come to {.0625 .3125 .6875 .9375 1} when you run it. These are all the probabilities we get when you replace "at most 2 girls" with "at most 0", "at most 1", etc. Here, .0625 is the probability of "at most 0" girls (or just 0 girls), .3125 is the probability of at most 1 girl (1 or 0 girls), etc.

Several other probability problems actually are the same as this one. For example, "less than N" girls, just means "at most N-1" girls. "At least N" girls means "at most (total-N)" boys (here we switch our definition of what a success is). "No more than", of course, means the same as "at most".

= Advanced (for programmers) =

The [[TI-BASIC:Binompdf|Binompdf(]] and binomcdf( commands are the only ones apart from [[TI-BASIC:Seq_List|seq(]] that can return a list of a given length, and they do it much more quickly. It therefore makes sense, in some situations, to use these commands as substitutes for seq(.

Here's how to do it:

# cumSum(binomcdf(N,0 gives the list {1 2 ... N+1}, and cumSum(not(binompdf(N,0 gives the list {0 1 2 ... N}.
# With seq(, you normally do math inside the list: seq(3I^^2^^,I,0,5
# With these commands, you do the same math outside the list: 3Ans^^2^^ where Ans is the list {0 1 ... 5}.

:seq(2^I,I,1,5
can be
:cumSum(binomcdf(4,0
:2^Ans
which in turn can be
:2^cumSum(binomcdf(4,0

In general (where f() is some operation or even several operations):

:seq(f(I),I,1,N
can be
:cumSum(binomcdf(N-1,0
:f(Ans
which can sometimes be
:f(cumSum(binomcdf(N-1,0

If the lower bound on I in the seq( statement is 0 and not 1, you can use binompdf( instead:

:seq(f(I),I,0,N
can be
:cumSum(not(binompdf(N,0
:f(Ans
which can sometimes be
:f(cumSum(not(binompdf(N,0

This will not work if some command inside seq( can take only a number and not a list as an argument. For example, seq(L,,1,,(I),I,1,5 cannot be optimized this way.

= Formulas =

Since "at most N" is equivalent to "0 or 1 or 2 or 3 or ... N", and since we can combine these probabilities by adding them, we can come up with an expression for binomcdf( by adding up values of [[TI-BASIC:Binompdf|Binompdf(]]:

[[TI-BASIC:Math|Math]]
\operatorname{binomcdf}(n,p,k) = \sum_{i=0}^{k}\operatorname{binompdf}(n,p,i) = \sum_{i=0}^{k}\binom{n}{i}\,p^i\,(1-p)^{n-i}
[[TI-BASIC:/math|/math]]

(If you're not familiar with sigma notation, <math>\sum_{i=0}^{k}</math> just means "add the following up for each value of i 0 through k")

= Error Conditions =

* '''[[TI-BASIC:Errors#datatype|ERR:DATATYPE]]''' is thrown if you try to generate a list of probabilities with ''p'' equal to 0 or 1, and at least 257 trials.
* '''[[TI-BASIC:Errors#domain|ERR:DOMAIN]]''' is thrown if the number of trials is at least 1 000 000, unless the other arguments make the problem trivial.

= Related Commands =

* [[TI-BASIC:Binompdf|Binompdf(]]
* [[TI-BASIC:Geometpdf|Geometpdf(]]
* [[TI-BASIC:Geometcdf|Geometcdf(]][[Category:TI-BASIC]]
[[Category:TIBD]]

TI-BASIC:Binomcdf - Revision history

Maintenance script: Automated superscript correction

KermMartian at 19:25, 24 February 2016

Maintenance script: Initial automated import