Alex's copy of Find CDF of given exponential distribution, and expectation and variance,

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Alex Van den Hof

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Alex Van den Hof 8 years ago

Created this as a copy of Find CDF of given exponential distribution, and expectation and variance, .

Name	Status	Author	Last Modified
Find CDF of given exponential distribution, and expectation and variance,	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Alex's copy of Find CDF of given exponential distribution, and expectation and variance,	draft	Alex Van den Hof	31/03/2017 11:40

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Question statement

Suppose $Y \sim \operatorname{Exp}(\var{la})$ with PDF $f_Y(y)=\var{la}e^{-\var{la}y},\;\;y \gt 0$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	number	4.5
b	number	7.5
la	number	1
ans2	number	0.0106
tol	number	0.0002
tans2	number	0.0105559122

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Name

Data type

Value

xxxxxxxxxx
 
if(la>=4,random(0.5..1#0.5),if(la>2,random(0.5..3#0.5),random(0.5..5#0.5)))

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Description

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

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Generated value: `number`

4.5

← Depends on:

→ Used by:

tans2

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
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Part c) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Gap-fill

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Find the CDF of $Y$ .

$F_Y(y)=\;$ ?Unnamed gap, $y \gt 0$ .

Advice

We have

$\begin{eqnarray*} F_Y(y)&=&P(Y\le y)\\&=&\int_0^y\var{la}e^{-\var{la}x}dx\\&=&\left[-e^{-\var{la}x}\right]_0^y=1-e^{-\var{la}y}\end{eqnarray*}$

$\begin{eqnarray*}P(\var{a} \leq Y \leq \var{b})&=&F_Y(\var{b})-F_Y(\var{a})\\&=&1-e^{-\var{la}\times \var{b}}-(1-e^{-\var{la}\times \var{a}})\\&=&e^{-\var{la}\times \var{a}}-e^{-\var{la}\times \var{b}}\\&=&\var{tans2}=\var{ans2}\end{eqnarray*}$ to 4 decimal places.

The properties of the exponential distribution give:

$\displaystyle \operatorname{E}[Y]=\simplify[all,fractionNumbers]{1/{la}=2/{2*la}}$

$\displaystyle \operatorname{Var}(Y)=\simplify[all,fractionNumbers]{1/{la}^2=4/{4*la^2}}$

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This question is used in the following exams:

Chapter X: probability by Alex Van den Hof in LeicesterPhysPractice.
Simon's copy of Chapter X: probability by Simon Vaughan in Simon's workspace.
Unit 5 Core Exercises by Simon Vaughan in PA1710.

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Alex's copy of Find CDF of given exponential distribution, and expectation and variance,

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Taxonomy: mathcentre

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Alex Van den Hof 8 years ago

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Generated value: `number`

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Alex's copy of Find CDF of given exponential distribution, and expectation and variance,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Alex Van den Hof 8 years ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`