Find expectation, variance and CDF of uniform 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

Let $Y$ be a random variable with the uniform distribution

$Y \sim \operatorname{U}(\var{lower},\var{upper})$

Give any introductory information the student needs.

			Name	Type	Generated Value

upper	integer	2
lower	integer	-4
ans3	decimal	dec("0.500")
d	decimal	dec("-3")
f1	decimal	dec("-3")
ans1	rational	-1
ans2	number	3.000
f	decimal	dec("-1")
ans4	decimal	dec("0.333")
c	decimal	dec("-1")
tol	number	0.001
d1	decimal	dec("-1")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
lower+random(2..20#2)

JME function reference

Description

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

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Generated value: `integer`

← Depends on:

lower

→ Used by:

ans1
ans2
ans3
ans4
c
d1
f1

This variable doesn't seem to be used anywhere.

Parts

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

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The expectation $\operatorname{E}[Y]=\;?$ Gap 0. (to 3 decimal places).

The variance $\operatorname{Var}(Y)=\;$ ?Gap 1. (to 3 decimal places).

Advice

a) For a Uniform distribution

$Y \sim \operatorname{U}(\var{lower},\var{upper})$ we have:

$\displaystyle \operatorname{E}[Y] = \simplify[!collectNumbers]{({lower}+{upper})/2}=\var{ans1}$

$\displaystyle \operatorname{Var}(Y) =\simplify[basic,!collectNumbers,!noleadingminus]{({upper}-{lower})^2/12}=\frac{\var{upper-lower}^2}{12}=\var{ans2}$ to 3 decimal places.

b) The CDF for a uniform distribution $Y$ on the interval $a \le y \le b$ is given by:

$F_Y(y) = \begin{cases} 0 & y \lt a \\ \frac{y-a}{b-a} & a \le y \le b, \\ 1 & y \gt b. \end{cases}$

Hence in this case we have:

$F_Y(y) = \simplify[basic]{(y-{lower})/({upper-lower})}$ for

$\var{lower}\le y \le \var{upper}$

c) Using the CDF we have:

$\begin{eqnarray}P(\var{d} \lt Y \lt \var{f})&=&F_Y(\var{f})-F_Y(\var{d})\\&=& \simplify[basic]{({f}-{lower})/({upper-lower})- ({d}-{lower})/({upper-lower})}\\&=&\var{ans4}\end{eqnarray}$

to 3 decimal places.

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Find expectation, variance and CDF of uniform 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: `integer`

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Find expectation, variance and CDF of uniform 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: integer

Parts

Gap-fill

Generated value: `integer`