Probability, expectation and standard deviation of binomial distribution

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Probability, expectation and standard deviation of binomial distribution.

Name	Status	Author	Last Modified
Probability, expectation and standard deviation of binomial distribution	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Probability, expectation and standard deviation of binomial distribution	draft	Xiaodan Leng	10/07/2019 22:31

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Question statement

{pre} $\var{thismany}$ {post}

{something} $\var{number1}$ {else}

Give any introductory information the student needs.

			Name	Type	Generated Value

pre	string
descx1	string	number of chocolate chip cooki
something	string
thisnumber	integer	6
what	string	daily sales.
things	string	chocolate chip cookies.
descx	string	the number of chocolate chip c
tol	number	0.001
prob	rational	17/100
thisaswell	string	our selection contains no more
else	string	biscuits are selected at rando
thismany	integer	17
number1	integer	8
post	string	% of biscuits made by a baker
prob2	number	0.594
prob1	number	0.000
thatnumber	integer	1
this	string	our selection contains exactly
v	integer	0
tprob1	number	0.0004655944
tprob2	number	0.5942795167
sd	decimal	dec("1.062")

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Description

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Generated value: `string`

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Statement

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Assuming a binomial distribution for $X$ , {descX}, write down the values of $n$ and $p$.

$X \sim \operatorname{bin}(n,p)$

$n=\; $?Gap 0. $p=\;$?Gap 1.

Find $\operatorname{E}[X]$ the expected {descX1}

$\operatorname{E}[X]=$?Gap 2.

Find the standard deviation for the {descX1}

Standard deviation = ? Gap 3. (to 3 decimal places).

Advice

1. $X \sim \operatorname{bin}(\var{number1},\var{prob})$, so $n= \var{number1},\;\;p=\var{prob}$.

2. The expectation is given by $\operatorname{E}[X]=n\times p=\var{number1}\times \var{prob}=\var{number1*prob}$

3. $\operatorname{stdev}(X)=\sqrt{n\times p \times (1-p)}=\sqrt{\var{number1}\times \var{prob} \times \var{1-prob}}=\var{sd}$ to 3 decimal places.

1. \[ \begin{eqnarray*}\operatorname{P}(X = \var{thisnumber}) &=& \dbinom{\var{number1}}{\var{thisnumber}}\times\var{prob}^{\var{thisnumber}}\times(1-\var{prob})^{\var{number1-thisnumber}}\\& =& \var{comb(number1,thisnumber)} \times\var{prob}^{\var{thisnumber}}\times\var{1-prob}^{\var{number1-thisnumber}}\\&=&\var{prob1}\end{eqnarray*} \] to 3 decimal places.

\[ \begin{eqnarray*}\operatorname{P}(X \leq \var{thatnumber})& =& \simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\& =& \simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})}\\& =& \var{prob2}\end{eqnarray*} \]

to 3 decimal places.

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Probability, expectation and standard deviation of binomial distribution

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Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

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Generated value: `string`

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Gap-fill

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Probability, expectation and standard deviation of binomial distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

Generated value: `string`