Probability, expectation and standard deviation of Poisson distribution

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

{pre} $\var{thismany}$ .

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

Give any introductory information the student needs.

			Name	Type	Generated Value

pre	string	The mean number of sales per d
what	string	daily sales.
this	string	a randomly selected employee m
things	string	sales.
number1	integer	3
descx	string	the number of sales per day
else	string	per day.
thismany	integer	9
something	string	Employees receive a warning if
tol	number	0.001
v	integer	1
tprob1	number	0.091090319
sd	number	3.000
tprob2	number	0.0062321951
prob2	number	0.006
thisnumber	integer	6
thisaswell	string	a randomly selected employee r
prob1	number	0.091

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Name

Data type

Value

xxxxxxxxxx
 
"The mean number of sales per day at a telecommunications centre is "

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `string`

The mean number of sales per day at a telecommunications centre is

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

Gap-fill

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Assuming a Poisson distribution for $X$ , {descX}, write down the value of $\lambda$ .

$X \sim \operatorname{Poisson}(\lambda)$

$\lambda =$ ?Gap 0.

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

$\operatorname{E}[X]=$ ?Gap 1.

Find the standard deviation for {what}.

Standard deviation = ? Gap 2. (to 3 decimal places).

Advice

1. $X \sim \operatorname{Poisson}(\var{thismany})$ , so $\lambda = \var{thismany}$ .

2. The expectation is given by $\operatorname{E}[X]=\lambda=\var{thismany}$

3. $\operatorname{stdev}(X)=\sqrt{\lambda}=\sqrt{\var{thismany}}=\var{sd}$ to 3 decimal places.

$\begin{eqnarray*}\operatorname{P}(X = \var{thisnumber}) &=& \frac{e ^ { -\var{thismany}}\var{thismany} ^ {\var{thisnumber}}} {\var{thisnumber}!}\\& =& \var{prob1} \end{eqnarray*}$ to 3 decimal places.

2. If an employee receives a warning then he or she must have sold less than {number1}.

Hence we need to find :

$\begin{eqnarray*}\operatorname{P}(X < \var{number1})& =& \simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\& =& \simplify[all,!collectNumbers]{e ^ { -thismany} + {thismany} * e ^ { -thismany} + {v} * (({thismany} ^ 2 * e ^ { -thismany}) / 2)} \\&=& \var{prob2} \end{eqnarray*}$

to 3 decimal places.

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Probability, expectation and standard deviation of Poisson distribution

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `string`

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Probability, expectation and standard deviation of Poisson distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

Generated value: `string`