Find maximum likelihood estimator of mean of normal distribution

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Bill Foster 6 years, 4 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

In an experiment we take $\var{n}$ observations $x_i$ from a Normal distribution with unknown mean $\mu$ and variance $\var{var}$.

From this sample we find:

\[\sum_{i=1}^{\var{n}}x_i = \var{m}\]

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	11
b	integer	2
d	integer	5
g	integer	3
m	integer	305
n	integer	20
w	integer	0
mlemu	decimal	dec("15.25")
u	integer	0
t	integer	0
tol	integer	0
v	integer	1
var	integer	27
r	integer	3
mlephi	decimal	dec("4.00")

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Name

Data type

Value

xxxxxxxxxx
 
random(2..9)

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Description

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Number entry
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Part b) Gap-fill
1. Gap Number entry
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Gap-fill

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Using this data find the MLE $\hat{\mu}$ of $\mu$ .

$\hat{\mu}=\;\;$Unnamed gap

Enter to 2 decimal places.

Advice

a)
The MLE $\hat{\mu}$ of $\mu$ is given by dividing the sum of the sample by the number of samples i.e.

\[\hat{\mu}=\frac{\var{m}}{\var{n}}=\var{mlemu}\] to 2 decimal places.

b)
In order to find the MLE $\hat{\phi}$ of \[\phi=\simplify[std]{{t}*ln(mu+{b})+{u}*({a}/(mu+{d}))+{v}e^(mu/{c})+{w}*(mu^2+{g})}\]

we note that $\phi$ is 1-1 and so we can directly substitute $\mu=\hat{\mu}$ into this to obtain:

\[\hat{\phi} = \simplify[std]{{t} * Ln({m} / {n} + {b}) + {u} * ({a} / ({m} / {n} + {d})) + {v} * e ^ (({m} / {n}) / {c}) + {w} * (({m} / {n}) ^ 2 + {g})} = \var{mlephi}\]

to 2 decimal places.

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Find maximum likelihood estimator of mean of normal distribution

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Bill Foster 6 years, 4 months ago

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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Gap-fill

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Find maximum likelihood estimator of mean of normal distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 4 months ago

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`