Geometric Distribution

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Brad Allison

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5 years, 8 months ago

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Brad Allison 5 years, 7 months ago

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Brad Allison 5 years, 8 months ago

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Brad Allison 5 years, 9 months ago

Gave some feedback: Needs to be tested

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Brad Allison 5 years, 9 months ago

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There is only one version of this question that you have access to.

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Question statement

This quiz tests the calculations for Geometric probabilities. Before you start you may wish to note down the probability mass function and cumulative density function formulae to help answer the questions.

Give any introductory information the student needs.

			Name	Type	Generated Value

p1	number	0.45
r1	integer	2
Ans1	number	0.2475
r2	integer	5
Ans2	number	0.0411778125
r3	integer	10
Ans3	number	0.002072415
p2	number	0.45
r4	integer	2
Ans4	number	0.3025
r5	integer	3
Ans5	number	0.833625
r6	integer	4
Ans6	number	0.833625
r7	integer	5
j6	integer	4
Ans7	number	0.09150625
j1	integer	3

			Name	Type	Generated Value

q	integer	6
p3	rational	1/6
n1	integer	4
Ans8	number	0.0964506173
n2	integer	2
Ans9	number	0.3055555556
n3	integer	7
j3	integer	6
Ans10	number	0.3348979767
n4	integer	9
Ans11	number	0.8061933005
q2	integer	25
p4	rational	1/4
n5	integer	5
Ans12	number	0.0791015625
n6	integer	7
Ans13	number	0.8665161133
n7	integer	3
Ans14	number	0.5625
j4	integer	2

			Name	Type	Generated Value

f1	integer	36
p5	rational	1/36
n8	integer	13
j5	integer	12
Ans15	number	0.7131592556
Ans16	rational	36

			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: `number`

0.45

This variable doesn't seem to be used anywhere.

Parts

Question 1 Gap-fill
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Question 3 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 $X \sim Geo(\var{p1})$ . Find,

(i) $P(X=\var{r1})$

Answer: Gap 0.

(ii) $P(X=\var{r2})$

Answer: Gap 1.

(iii) $P(X=\var{r3})$

Answer: Gap 2.

(b) Given that $X \sim Geo(\var{p2})$ . Find,

(i) $P(X>\var{r4})$

Answer:Gap 3.

(ii) $P(X\leq\var{r5})$

Answer:Gap 4.

(iii) $P(X<\var{r6})$

Answer:Gap 5.

(iv) $P(X\geq\var{r7})$

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: $X\sim Geo(\var{P1})$

(i) $P(X= x)=p(1-p)^{x-1} \rightarrow P(X=\var{r1}) = \var{p1}(1-\var{p1})^{\var{r1}-1} = \var{Ans1}$

(ii) $P(X= x)=p(1-p)^{x-1} \rightarrow P(X=\var{r2}) = \var{p1}(1-\var{p1})^{\var{r2}-1} = \var{Ans2}$

(iii) $P(X= x)=p(1-p)^{x-1} \rightarrow P(X=\var{r3}) = \var{p1}(1-\var{p1})^{\var{r3}-1} = \var{Ans3}$

1b: $X\sim Geo(\var{P2})$

(i) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X>\var{r4}) = 1-P(X\leq\var{r4}) = 1-(1-(1-\var{p2})^{\var{r4}})=\var{Ans4}$

(ii) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\leq\var{r5}) = 1-(1-\var{p2})^{\var{r5}}=\var{Ans5}$

(iii) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X<\var{r6}) = P(X\leq \var{r6}-1)= P(X\leq \var{j1})= 1-(1-\var{p2})^\var{j1}=\var{Ans6}$

(iv) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\geq\var{r7}) =1-P(X\leq \var{j6}) =1-(1-(1-p)^\var{j6})=\var{Ans7}$

Question 2

2a: $ X\sim Geo(\var{P3})$

(i) $P(X= x)=p(1-p)^{x-1} \rightarrow P(X=\var{n1}) = \frac{1}{\var{q}}(1-\frac{1}{\var{q}})^{\var{n1}-1} = \var{Ans8}$

(ii) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X=\var{n2})= 1-(1-\frac{1}{\var{q}})^{\var{n2}-1} = \var{Ans9}$

(iii) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\geq \var{n3}) = 1-P(X\leq \var{n3}-1) = 1-P(X\leq\var{j3}) = 1-(1-(1-p)^\var{j3})=\var{Ans10}$

(iv) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\leq \var{n4}) = 1-(1-p)^\var{n4}=\var{Ans11}$

2b: $ X\sim Geo(\var{P4})$

(i) $P(X= x)=p(1-p)^{x-1} \rightarrow P(X=\var{n5}) = \var{p4}(1-\var{p4})^{\var{n5}-1} = \var{Ans12}$

(ii) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\leq \var{n6}) = 1-(1-\var{p4})^{\var{n6}} = \var{Ans13}$

(iii) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\geq \var{n7}) = 1-P(X\leq\var{n7}-1)=1-P(X\leq \var{j4})=1-(1-(1-\var{p4})^\var{j4}) = \var{Ans14}$

Question 3

3a: $X\sim Geo(\var{p5})$

(i) $P(X\leq x)=1-(1-p)^{\mid x\mid} \rightarrow P(X\geq \var{n8}) = 1-P(X\leq\var{n8}-1)=1-P(X\leq \var{j5})=1-(1-(1-\var{p5})^{\var{j5}}) = \var{Ans15}$

(ii) $E(X)=\frac{1}{p} =\frac{1}{\frac{1}{\var{f1}}} = \var{Ans16}$

Give a worked solution to the whole question.

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This question is used in the following exams:

Week 3 test by Ellen Marshall in Ellen's workspace.
Geometric Distribution by Brad Allison in Exam Version of Tests.
Xiaochuan's copy of Discrete random variables by Xiaochuan Yang in Xiaochuan's workspace.
Discrete random variables by Anne-Sophie Kaloghiros in MA1670.
Long Test by Anne-Sophie Kaloghiros in MA1670.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Brad Allison 5 years, 7 months ago

Brad Allison 5 years, 8 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: `number`

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Geometric Distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Brad Allison 5 years, 7 months ago

Brad Allison 5 years, 8 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: number

Parts

Gap-fill

Generated value: `number`