The probability $P(X \\ge \\var{x1})$ is given by:
\n\\[P(X\\ge \\var{x1})= \\int_a^b \\int_c^d f(x,y)dxdy\\]
\nInput the limits of integration here:
\n$a=\\;$?[[0]] $b=\\;$?[[1]]
\n$c=\\;$?[[2]] $d=\\;$?[[3]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{c}", "minValue": "{c}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{c+d}", "minValue": "{c+d}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a}", "minValue": "{a}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+b}", "minValue": "{a+b}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\\[f_X(x)=\\int_p^qf(x,y)dy\\]
\nInput the limits of integration here:
\n$p=\\;$?[[0]] $q=\\;$?[[1]]
\nAlso input the range of values of $x,\\;x_1 \\le x \\le x_2$ on which $f_X(x)$ is defined.
\n$x_1=\\;$?[[2]] $x_2=\\;$?[[3]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a}", "minValue": "{a}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+b}", "minValue": "{a+b}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{c}", "minValue": "{c}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{c+d}", "minValue": "{c+d}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "The marginal $f_Y(y$) of $Y$ is calculated as follows: \\[f_Y(y)=\\int_r^s f(x,y)dx\\]
\nInput the limits of integration here:
\n$r=\\;$?[[0]] $s=\\;$?[[1]]
\nAlso input the range of values of $y,\\;y_1 \\le y \\le y_2$ that $f_Y(y)$ is defined on.
\n$y_1=\\;$?[[2]] $y_2=\\;$?[[3]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{c}", "minValue": "{c}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{y1}", "minValue": "{y1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{x1}", "minValue": "{x1}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a+b}", "minValue": "{a+b}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "The conditional probability $P(Y \\le \\var{y1}|X \\ge \\var{x1})$ is calculated as follows:
\n\\[P(Y \\le \\var{y1}|X \\ge \\var{x1})=\\frac{\\int_t^u\\int_v^w f(x,y) dx dy}{A}\\]
\nWhere:
\n\\[A = \\int_{\\var{c}}^{\\var{c+d}}\\int_{\\var{x1}}^{\\var{a+b}} f(x,y) dx dy\\]
\nInput the limits of integration here:
\n$t=\\;$?[[0]] $u=\\;$?[[1]]
\n$v=\\;$?[[2]] $w=\\;$?[[3]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "Suppose that a bivariate continuous random variable is defined on the region $R \\subset \\mathbb{R^2}$ given by:
\n\\[\\var{a} \\le x \\le \\var{a+b},\\;\\;\\;\\;\\;\\;\\;\\var{c} \\le y \\le \\var{c+d}\\] with joint PDF given by $f(x,y)$ in $R$ and zero otherwise.
\nIn the following questions, you are asked to supply the limits of integration for the following computations:
", "tags": ["bivariate distributions", "checked2015", "conditional probability", "double integration", "limits of integration", "marginal distributions", "MAS1604", "MAS2304", "pdf", "PDF", "Probability", "statistics"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "6/02/2013:
\nFinished first draft.
", "licence": "Creative Commons Attribution 4.0 International", "description": "$f(x,y)$ is the PDF of a bivariate distribution $(X,Y)$ on a given rectangular region in $\\mathbb{R}^2$. Write down the limits of the integrations needed to find $P(X \\ge a)$, the marginal distributions $f_X(x),\\;f_Y(y)$ and the conditional probability $P(Y \\le b|X \\ge c)$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nThe probability $P(x \\ge \\var{x1})$ is given by:
\n\\[P(x \\ge \\var{x1})=\\int_{\\var{c}}^{\\var{c+d}}\\int_{\\var{x1}}^{\\var{a+b}} f(x,y) dx dy\\]
\nb)
\nThe marginal $f_X(x)$ of $X$ is:
\n\\[f_X(x)=\\int_{\\var{c}}^{\\var{c+d}}f(x,y)dy\\] and the range that $f_X(x)$ is defined on is $\\var{a} \\le x \\le \\var{a+b}$.
\nc)
\nThe marginal $f_Y(y)$ of $Y$ is:
\n\\[f_Y(y)=\\int_{\\var{a}}^{\\var{a+b}}f(x,y)dx\\] and the range that $f_Y(y)$ is defined on is $\\var{c} \\le x \\le \\var{c+d}$.
\nd) The conditional probability $P(Y \\le \\var{y1}|X \\ge \\var{x1})$ is calculated as follows:
\n\\[P(Y \\le \\var{y1}|X \\ge \\var{x1})=\\frac{\\int_{\\var{c}}^{\\var{y1}}\\int_{\\var{x1}}^{\\var{a+b}} f(x,y) dx dy}{A}\\]
\n ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}