Poisson Distribution

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

x1	integer	3
hrs	rational	3/2
mins	integer	90
Ans1	number	0.0892350784
rate	integer	4
Lambda1	rational	6
xx1	integer	4
Ansxx	number	0.1338526175
rate2	integer	6

			Name	Type	Generated Value

			Ans2	number	0.9380311956
			x2	integer	3

			Name	Type	Generated Value

Timemins	integer	15
Timeminsno	rational	25
Timehrs	integer	13
Daypart	string	pm
Totaltime	rational	53/4
Lambda2	rational	9
Ans3	rational	135
Hours	rational	9/4
x3	integer	0
Totalmins	rational	135
Ans4	number	0.9998765902

			Name	Type	Generated Value

Ans5	number	0.2700164243
Ans6	number	0.0992310359
Ans7	number	0.0416770351
Lambda3	number	2.1
x4	integer	1
x5	integer	2
x6	integer	4
x7	integer	5

			Name	Type	Generated Value

Lambda4	number	2.9
x8	integer	3
x9	integer	4
x10	integer	5
x11	integer	6
Ans8	number	0.4459631986
Ans9	number	0.8317770763
Ans10	number	0.0741738016
Ans11	number	0.0741738016

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

A random number between and (inclusive) with step size

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

Ans1

Advice
"Question 2" - prompt

Parts

Question 1 Gap-fill
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Gap-fill

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

(a) Given that $X\sim Po\left({\var{Lambda3}}\right)$ . Find,

(i) $P\left(X=\var{x5}\right)$

Answer:Gap 0.

(ii) $P\left(X=\var{x6}\right)$

Answer:Gap 1.

(iii) $P\left(X=\var{x7}\right)$

Answer:Gap 2.

(b) Given that $X\sim Po\left({\var{Lambda4}}\right)$ . Find,

(i) $P(X<\var{x8})$

Answer:Gap 3.

(ii) $P(X\le\var{x9})$

Answer:Gap 4.

(iii) $P(X>\var{x10})$

Answer:Gap 5.

(iv) $P(X\ge\var{x11})$

Answer:Gap 6.

Advice

Here are some tips on how to set out your work, as well as how to complete the questions. None of the answers given below have been rounded to what the question states.

Question 1

1a: $X\sim Po(\var{Lambda3})$

(i) $P(X=x)=P(X=\var{x5})=\frac{e^{-\var{Lambda3}}\times\var{Lambda3}^{\var{x5}}}{\var{x5}!}=\var{Ans5}$

(ii) $P(X=x)=P(X=\var{x6})=\frac{e^{-\var{Lambda3}}\times\var{Lambda3}^{\var{x6}}}{\var{x6}!}=\var{Ans6}$

(iii) $P(X=x)=P(X=\var{x7})=\frac{e^{-\var{Lambda3}}\times\var{Lambda3}^{\var{x7}}}{\var{x7}!}=\var{Ans7}$

1b: $X\sim Po(\var{Lambda4})$

(i) $P(X=x)=P(X<\var{x8})=P(X=0)+P(X=1)+P(X=2)=\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^0} {0!}+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^1} {1!}+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^2} {2!}=\var{Ans8}$

(ii) $P(X=x)=P(X\leq\var{x9})=P(X=0)+P(X=1)+...+P(X={\var{x9}})=\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^0} {0!}+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^1} {1!}+...+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^{\var{x9}}} {\var{x9}!}=\var{Ans9}$

(iii) $P(X=x)=P(X>\var{x10})=1-P(X\leq\var{x10})=1-(P(X=0)+P(X=1)+...+P(X={\var{x10}}))=1-(\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^0} {0!}+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^1} {1!}+...+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^{\var{x10}}} {\var{x10}!})=\var{Ans10}$

(iv) $P(X=x)=P(X\geq\var{x11})=1-P(X<\var{x11})=1-(P(X=0)+P(X=1)+...+P(X={5}))=1-(\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^0} {0!}+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^1} {1!}+...+\frac{e^{-\var{Lambda4}}\times\var{Lambda4}^{5}} {5!})=\var{Ans11}$

Question 2

2a: $\lambda=rate \times hours = \var{rate2} \times 1 = \var{rate2} \therefore X\sim Po(\var{rate2})$

(i) $P(X=x) = P(X=\var{xx1})=\frac{e^{-\var{rate2}}\times\var{rate2}^\var{xx1}}{\var{xx1}!}=\var{Ansxx}$

2b: $\lambda=rate \times hours =rate \times \frac{mins}{60} = \var{rate}\times\frac{\var{mins}}{60}=\var{Lambda1} \therefore X\sim Po(\var{Lambda1})$

(i) $P(X=x) = P(X=\var{x1})=\frac{e^{-\var{Lambda1}}\times\var{Lambda1}^\var{x1}}{\var{x1}!}=\var{Ans1}$

(ii) $P(X=x) = P(X\geq\var{x2})=1-P(X<\var{x2})=1-(P(X=0)+P(X=1)+...+P(X={3}))=1-(\frac{e^{-\var{Lambda1}}\times\var{Lambda1}^0} {0!}+\frac{e^{-\var{Lambda1}}\times\var{Lambda1}^1} {1!}+...+\frac{e^{-\var{Lambda1}}\times\var{Lambda1}^{3}} {3!})=\var{Ans2}$

2c: $\lambda=rate \times hours=rate \times \frac{mins}{60} = \var{rate}\times\frac{\var{Totalmins}}{60}=\var{Lambda1} \therefore X\sim Po(\var{Lambda1})$

(i) This is the time difference in minutes, that the rate of patients arrives within.

(ii) $P(X=x) = P(X\geq1)=1-P(X<1)=1-P(X=0)=1-\frac{e^{-\var{Lambda1}}\times\var{Lambda1}^0} {0!}=\var{Ans4}$

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Taxonomy: Context

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Brad Allison 5 years, 8 months ago

Brad Allison 5 years, 8 months ago

Brad Allison 5 years, 9 months ago

Brad Allison 5 years, 9 months ago

Brad Allison 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Poisson Distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Brad Allison 5 years, 8 months ago

Brad Allison 5 years, 8 months ago

Brad Allison 5 years, 9 months ago

Brad Allison 5 years, 9 months ago

Brad Allison 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`