AP Statistics Curriculum 2007 Distrib Binomial - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T17:46:35Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Binomial Expectation and Variance */

2014-09-17T22:25:12Z

‎Binomial Expectation and Variance

IvoDinov: /* Binomial Coin Toss Example */ corected a typo on this formula ( $E[X]=\sum_x{xP(X=x)} =$ =0\times{5\choose 0}0.3^0(0.7)^{5}+1\times{5\choose 1}0.3^1(0.7)^{4}+2\times ...)

2011-02-02T18:21:01Z

‎Binomial Coin Toss Example: corected a typo on this formula (<math>E[X]=\sum_x{xP(X=x)} = </math> =0\times{5\choose 0}0.3^0(0.7)^{5}+1\times{5\choose 1}0.3^1(0.7)^{4}+2\times ...)

Jenny: /* Binomial Coin Toss Example */

2010-06-28T19:50:00Z

‎Binomial Coin Toss Example

Jenny: /* Examples */

2010-06-28T19:48:45Z

‎Examples

Jenny: /* Approximate Binomial Model */

2010-06-28T19:48:14Z

‎Approximate Binomial Model

Jenny: /* Bernoulli process */

2010-06-28T19:46:57Z

‎Bernoulli process

Jenny at 16:58, 28 June 2010

2010-06-28T16:58:54Z

IvoDinov: added a link to the Problems set

2009-10-26T04:53:29Z

added a link to the Problems set

IvoDinov: /* Bernoulli process */ typo

2008-04-30T17:06:19Z

‎Bernoulli process: typo

← Older revision		Revision as of 17:46, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Binomial}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Distrib_Binomial}}

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 ===Binomial Expectation and Variance===
 If X is the random variable representing the r(random) number of heads in n coin toss trials, where the P(Head) = p, i.e., <math>X\sim B(n, p)</math>. Then we have the following expressions for the [[AP_Statistics_Curriculum_2007#Expectation_.28Mean.29_and_Variance | expected value, variance and the standard deviation]] of X :
-* Mean (expected value): <math>E[X] = n\times p</math>,
+* Mean (expected value): <math>E[X] = n\times p,</math>
-* Variance: <math>VAR[X] = n\times p \times(1-p)</math>, and
+* Variance: <math>VAR[X] = n\times p \times(1-p),</math>
-* Standard deviation: <math>SD[X] = \sqrt{n\times p \times(1-p)}</math>
+* Standard deviation: <math>SD[X] = \sqrt{n\times p \times(1-p)}.</math>
 * [http://en.wikipedia.org/wiki/Binomial_distribution The complete description of the Binomial distribution is available here]

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 * Theoretical calculation, using the Binomial Probabilities;<math>E[X]=\sum_x{xP(X=x)} = </math>
-<math>={5\choose 0}0.3^0(0.7)^{5}+{5\choose 1}0.3^1(0.7)^{4}+{5\choose 2}0.3^2(0.7)^{3}+{5\choose 3}0.3^3(0.7)^{2}+{5\choose 4}0.3^4(0.7)^{1}+{5\choose 5}0.3^5(0.7)^{0} =</math>
+<math>=0\times{5\choose 0}0.3^0(0.7)^{5}+1\times{5\choose 1}0.3^1(0.7)^{4}+2\times{5\choose 2}0.3^2(0.7)^{3}+3\times{5\choose 3}0.3^3(0.7)^{2}+4\times{5\choose 4}0.3^4(0.7)^{1}+5\times{5\choose 5}0.3^5(0.7)^{0} =</math>
 <math> \cdots = (n\times p) = 5\times 0.3 = 1.5.</math>

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 <center>[[Image:SOCR_EBook_Dinov_RV_Binomial_013008_Fig2.jpg|400px]]</center>
-* Notes: Of course, the theoretical calculation is exact and the empirical calculation is only approximate. However, the power of the empirical approximation to the expected number of Heads becomes clear when we increase the number of coins from 5 to 100, or more. In these cases, the exact calculations become very difficult, and for even larger number of coins become intractable.
+* Notes: Of course, the theoretical calculation is exact and the empirical calculation is only approximate. However, the power of the empirical approximation to the expected number of Heads becomes clearer when we increase the number of coins from 5 to 100, or more. In these cases, the exact calculations become very difficult, and for even larger number of coins become intractable.
 ===[[EBook_Problems_Distrib_Binomial |Problems]]===

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 <center>[[Image:SOCR_EBook_Dinov_RV_Binomial_013008_Fig1.jpg|400px]]</center>
-* Suppose 10% of the human population carries the green-eye allele. If we choose 1,000 people randomly and let the RV X be the number of green-eyed people in the sample. Then the distribution of X is binomial distribution with n = 1,000 and p = 0.1 (denoted as <math>X \sim B(1,000, 0.1)</math>. In a sample of 1,000, how many are we expecting to have this allele?
+* Suppose 10% of the human population carries the green-eye allele. If we choose 1,000 people randomly and let the RV X be the number of green-eyed people in the sample. Then the distribution of X is binomial distribution with n = 1,000 and p = 0.1 (denoted as <math>X \sim B(1,000, 0.1)</math>. In a sample of 1,000, how many of them are we expecting to have this allele?
 ===Binomial Modeling===

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 Suppose we observe an experiment comprised of n identical trials involving a large population (of size N). Assume the population contains a sub-population of subjects that have a characteristic of interest and the sub-population proportion is p (0<p<1). Then the distribution of X={the number of outcomes in the sample with that characteristic of interest}, is approximately Binomial(n, p). This approximation is adequate if the ratio n/N < 0.05.
-* Example: Polling the US population to see what proportion is/has-been married. Because we sample without replacement (can't ask the same individual twice), the second assumption of the (exact) Binomial model is (slightly) violated. Yet, the small ration of sample to population size yields that the Binomial model is approximately valid (i.e., the proportion of subjects that is/has-been married does not change significantly as we poll one subject, and therefore remove him/her from the complete pool of subjects we poll).
+* Example: Polling the US population to see what proportion is/has-been married. Since we sample without replacement (can't ask the same individual twice), the second assumption of the (exact) Binomial model is (slightly) violated. Yet, the small ration of sample to population size yields that the Binomial model is approximately valid (i.e., the proportion of subjects that is/has-been married does not change significantly as we poll one subject, and therefore remove him/her from the complete pool of subjects we poll).
 ===Binomial Probabilities===

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 * Examples of Bernoulli Trials
 ** A Coin Toss. We can obverse H="heads", conventionally denoted success, or T="tails" denoted as failure. A fair coin has the probability of success 0.5 by definition.
-** Rolling a Die: Where the outcome space is binarized to "success"={6} and "failure" = {1, 2, 3, 4, 5}.
+** Rolling a Die: The outcome space is binarized to "success"={6} and "failure" = {1, 2, 3, 4, 5}.
 ** Polls: Choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.

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 ===Examples===
 ====Binomial Coin Toss Example====
-Refer to the SOCR [[SOCR_EduMaterials_Activities_BinomialCoinExperiment | Binomial Coin Toss Experiment]] and use the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Binomial Coin Toss Applet] to perform an experiment of tossing a biased coin, P(Head) = 0.3, 5 times and computing the expectation of the number of Heads in such experiment. You should recognize that there are two distinct ways of computing the expected number of Heads:
+Refer to the SOCR [[SOCR_EduMaterials_Activities_BinomialCoinExperiment | Binomial Coin Toss Experiment]] and use the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Binomial Coin Toss Applet] to perform an experiment of tossing a biased coin, P(Head) = 0.3, 5 times and comput the expectation of the number of Heads in such experiment. You should recognize that there are two distinct ways of computing the expected number of Heads:
 * Theoretical calculation, using the Binomial Probabilities;<math>E[X]=\sum_x{xP(X=x)} = </math>

← Older revision		Revision as of 04:53, 26 October 2009
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	* Notes: Of course, the theoretical calculation is exact and the empirical calculation is only approximate. However, the power of the empirical approximation to the expected number of Heads becomes clear when we increase the number of coins from 5 to 100, or more. In these cases, the exact calculations become very difficult, and for even larger number of coins become intractable.		* Notes: Of course, the theoretical calculation is exact and the empirical calculation is only approximate. However, the power of the empirical approximation to the expected number of Heads becomes clear when we increase the number of coins from 5 to 100, or more. In these cases, the exact calculations become very difficult, and for even larger number of coins become intractable.
		+
		+	===[[EBook_Problems_Distrib_Binomial \|Problems]]===

	<hr>		<hr>

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 ** Polls: Choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.
-* The Bernoulli random variable (RV): Mathematically, a Bernoulli trial is modeled by a random variable <math>X(outcome) = \begin{cases}0,& s = \texttt{success},\\
+* The Bernoulli random variable (RV): Mathematically, a Bernoulli trial is modeled by a random variable <math>X(outcome) = \begin{cases}0,& s = \texttt{failure},\\
-,& s = \texttt{failure}.\end{cases}</math>
+,& s = \texttt{success}.\end{cases}</math>
 If p=P(success), then:
 : ''expected value'' of X, E[X]=p