// Numbas version: exam_results_page_options {"name": "Kma's copy of Log-likelihood and maximum likelihood estimator for PDF", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["prod", "mle", "m", "x1", "sumsq", "tol", "x3", "x2", "x", "y2", "where"], "name": "Kma's copy of Log-likelihood and maximum likelihood estimator for PDF", "tags": ["checked2015", "cr1", "density function", "estimators", "likelihood functions", "log-likelihood function", "maximum", "maximum likelihood estimator", "mle", "MLE", "PDF", "pdf", "Probability", "probability", "probability density function", "random sample", "random variable", "sc", "second derivative", "statistics", "tested1", "unused"], "type": "question", "advice": "

\\[ \\begin{eqnarray*} L(t|\\underline{x})&=& \\var{2*x1}te^{-\\var{x1}^2t}\\times \\var{2*x2}te^{-\\var{x2}^2t} \\times \\var{2*x3}te^{-\\var{x3}^2t}\\\\ &=& \\var{8*prod}t^3e^{-\\var{sumsq}t} \\end{eqnarray*} \\]
b)

The log-likelihood function is :
\\[\\begin{eqnarray*} l(t|\\underline{x})&=&\\ln\\left( \\var{8*prod}t^3e^{-\\var{sumsq}t}\\right)\\\\ &=&\\ln(\\var{8*prod})+3\\ln(t)-\\var{sumsq}t \\end{eqnarray*} \\]

We have:
\\[\\frac{\\partial\\;l}{\\partial\\;t}=\\frac{3}{t}-\\var{sumsq}\\]
Now:
\\[\\begin{eqnarray*} \\frac{\\partial\\;l}{\\partial\\;t}&=&0 \\\\ \\Rightarrow \\frac{3}{t}-\\var{sumsq}&=&0\\\\ \\Rightarrow t&=&\\frac{3}{\\var{sumsq}} = \\var{mle} \\end{eqnarray*} \\] to 3 decimal places.
And putting $t=\\hat{t}$ gives the MLE $\\hat{t}=\\var{mle}$

d)
\\[\\frac{\\partial^2\\;l}{\\partial\\;t^2}=-3t^{-2} \\lt 0\\]
when evaluated at any point including $t=\\hat{t}=\\var{mle}$.

Hence gives a maximum at $t=\\hat{t}$.

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n \n \n

Find the likelihood function for $t$ given these observations.

\n \n \n \n

$L(t|\\underline{x})=\\;\\;$[[0]]

\n \n ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({(8 *prod)} * (t ^ 3) * Exp(({( - sumsq)} * t)))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [-0.2, -0.1]}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

Hence find the log-likelihood function for $t$

$l(t|\\underline{x})=\\;\\;$[[0]]

If $\\ln(a)$, for some integer $a$, is a term in your answer, leave as $\\ln(a)$ and do not evaluate.

Find the MLE $\\hat{t}$ for $t$

$\\hat{t}=\\;\\;$[[0]]

Input to 2 decimal places.

", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "mle+tol", "minValue": "mle-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

Now verify that you have indeed found a maximum:

1. First find $\\displaystyle\\frac{\\partial^2\\;l}{\\partial\\;t^2}=\\;\\;$[[0]].

2. Using the value of the MLE to 2 decimal places you have found:

$\\displaystyle\\frac{\\partial^2\\;l}{\\partial\\;t^2}$ evaluated at $\\hat{t}$ = [[1]].

Input to 2 decimal places.

", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "-(3/t^2)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"allowFractions": false, "scripts": {}, "maxValue": "m+tol", "minValue": "m-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "preamble": {"css": "", "js": ""}, "statement": "

The average annual wind speed, $X$, at {where} has the following probability density function with parameter $t$ which you have to estimate:

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\end{array} \\right .$	$2txe^{-tx^2}$	$x \\gt 0,$

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

For three randomly selected years, we observe the following average wind speeds:

$x_1=\\var{x1},\\;\\;x_2=\\var{x2}$ and $x_3=\\var{x3}$.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variables": {"prod": {"definition": "x1*x2*x3", "templateType": "anything", "group": "Ungrouped variables", "name": "prod", "description": ""}, "mle": {"definition": "precround(3/sumsq,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "mle", "description": ""}, "m": {"definition": "-precround(3/mle^2,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "x1": {"definition": "random(2.5..5.5#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}, "sumsq": {"definition": "x1^2+x2^2+x3^2", "templateType": "anything", "group": "Ungrouped variables", "name": "sumsq", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "x3": {"definition": "random(2.5..7.5#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "x3", "description": ""}, "x2": {"definition": "if(y2=x1,y2+ random(-1..1#0.5),y2)", "templateType": "anything", "group": "Ungrouped variables", "name": "x2", "description": ""}, "x": {"definition": "x1+x2+x3", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}, "y2": {"definition": "random(3.5..6.5#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "y2", "description": ""}, "where": {"definition": "random('North Shields','Owslebury','Bradfield in West Yorkshire','Sheffield','Windy Nook', 'Hepple','Leeming','Linton-on-Ouse','Scampton','Cranwell','Keele','Bingley', 'Ecclefechan','Finningley','Foxrock, Dublin')", "templateType": "anything", "group": "Ungrouped variables", "name": "where", "description": ""}}, "showQuestionGroupNames": false, "metadata": {"notes": "

14/07/2012:

Added tags.

Corrected mistakes in Advice.

Added some text to make statement clearer re parameter $t$.

Rephrased questions in last question so that it is clear that the value to 2dps is used in the calculation.

Spaced Advice text.

New tolerance variable, tol=0 for last two questions.

Added line in prompt: If $\\ln(a)$, for some integer $a$, is a term in your answer, leave as $\\ln(a)$ and do not evaluate.

Improved display of correct answer in second question as $+\\;- $ together. Also improved correct answer display in second last question.

Important: set checking range between -0.2 and -0.1 rather than between 0 and 1 so that evaluation of likelihood function over the range does not suffer from underflow and incorrect answer marked as correct. This needs constant testing, have tested on bounday values and OK.

1/08/2012:

Added tags.

In the Advice section, moved \\Rightarrow to the beginning of the line instead of the end of the previous line.

21/12/2012:

Checked calculations, OK. Added tested1 tag.

Checked rounding, OK. Added cr1 tag.

Scenarios, so added sc tag.

", "description": "

Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\\hat{t}$ for $t$.

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