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Bill Foster

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Christian Lawson-Perfect

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10 years ago

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Christian Lawson-Perfect 10 years ago

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Bill Foster 12 years, 3 months ago

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Question statement

A random variable $X$ has a probability density function (PDF) given by:

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	0
valk	decimal	dec("0.0308")
xl	integer	4
p	integer	5
pval	decimal	dec("0.40")
xu	integer	9

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
0

JME function reference

Description

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

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Generated value: `integer`

→ Used by:

pval
valk

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
Part c) Gap-fill
1. Gap Mathematical expression
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Gap-fill

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$f_X(x) = \left \{ \begin{array}{l} \phantom{{.}} \\ \phantom{{.}} \\ \phantom{{.}} \end{array} \right .$	$kx$	$\var{xl} \leq x \leq \var{xu},$

	$0,$	$\textrm{otherwise.}$

What value of $k$ makes $f_X(x)$ into the pdf of a distribution?

Input your answer here as a fraction and not as a decimal.

$k=\;\;$Unnamed gap

Advice

a)

Note that in order for $f_X(x)$ to be a pdf it must satisfy two important conditions:

1. $f_X(x) \ge 0$ in the range $\var{xl} \le x \le \var{xu}$

2. The area under the curve given by $f_X(x)$ is $1$ and this implies that:
\[\int_{\var{xl}}^{\var{xu}}f_X(x)\;dx = 1\] as the value of the function is $0$ outside this range.

We first check condition 2. and then check that condition 1. is satisfied.

Note that \[\int kx\;dx = k\frac{x^2}{2}\] on forgetting the constant of integration.

Hence \[\begin{eqnarray*} \int_{\var{xl}}^{\var{xu}}kx\;dx &=&\frac{k}{2}(\var{xu}^2-\var{xl}^2)\\ &=&\frac{k}{2}\times \var{xu^2-xl^2} \end{eqnarray*} \]

But we must have this last value equal to $1$ hence:
\[ \frac{k}{2}\times \var{xu^2-xl^2}=1 \Rightarrow k = \simplify[std]{2/{xu^2-xl^2}}\]

Hence the pdf is:
\[f_X(x) = \simplify[std]{2/{xu^2-xl^2}x}\;\;\;\;\;\var{xl} \le x \le \var{xu}\]

We have to check condition 1. that the function $f_X(x)$ is positive for $\var{xl} \le x \le \var{xu} $ – but this is clear from
the definition of $f_X(x)$ and the value of $k$.

b)

If $F_X(x)$ is the distribution function of the distribution given by $f_X(x)$ then:

$F_X(x) = 0\;\;\;x \lt \var{xl},\;\;\;\;F_X(x)=1\;\;\;x \ge \var{xu}$

and for $\var{xl} \le x \le \var{xu}$:

\[\begin{eqnarray*} F_X(x)&=&\int_{-\infty}^x f_X(x)\;dx=\simplify[std]{2/{xu^2-xl^2}}\int_{\var{xl}}^x x\;dx\\ &=&\simplify[std]{2/{xu^2-xl^2}}\times\frac{\left(x^2-\var{xl}^2\right)}{2}\\ &=&\frac{x^2-\var{xl^2}}{\var{xu^2-xl^2}} \end{eqnarray*} \]

c)

We have
\[\begin{eqnarray*} P\left(X \lt \simplify[std]{{xl+xu}/2}\right)&=&F_X\left(\simplify[std]{{(xl+xu)}/2}\right)\\ &=& \frac{1}{\var{xu^2-xl^2}}\left(\simplify[std]{({(xl+xu)}/{2})^2-{xl}^2}\right)\\ &=&\simplify{{3*xl+xu-4*a}/{4*(xu+xl-2*a)}} \end{eqnarray*} \]

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This question is used in the following exams:

Introduction to Statistics 2 by Bill Foster in Bill's workspace.
Probability Density Functions by Daniel Organisciak in Daniel's workspace.
Maths Support: Probability density functions by Christian Lawson-Perfect in Maths Support Wiki.
mathcentre: Probability density functions by Newcastle University Mathematics and Statistics in mathcentre.
Probability: 1 by Aneela Arshad in Aneela's workspace.
Integration: 1 by Aneela Arshad in Aneela's workspace.
PDFs by Bill Foster in Bill's workspace.
Integration by Bill Foster in Bill's workspace.
Probability density functions by Simon Thomas in Stats.
Probability density functions by Richard Pymar in Richard's workspace.
mathcentre: Probability density functions by Michael Proudman in Mathcentre based Materials.
Jia's copy of mathcentre: Probability density functions by Jia Shao in Jia's workspace.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Christian Lawson-Perfect 10 years ago

Bill Foster 12 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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IS2.1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years ago

Bill Foster 12 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`