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e=localStorage.getItem(this.adaptiveId+this.divid+"Count");null!=e&&"NaN"!=e||(e=0),this.checkCount=e,this.recentAttempts=localStorage.getItem(this.adaptiveId+"recentAttempts"),null!=this.recentAttempts&&"NaN"!=this.recentAttempts||(this.recentAttempts=3),localStorage.setItem(this.adaptiveId+"recentAttempts",this.recentAttempts),localStorage.setItem(this.adaptiveId+"Problem",this.divid),localStorage.setItem(this.adaptiveId+this.divid+"Count",this.checkCount),localStorage.setItem(this.adaptiveId+"Solved",!1)}usesIndentation(){return!this.noindent&&0!=this.solutionIndent()}distractorToRemove(){var e,t=this.enabledAnswerBlocks();for(let i=0;i{t.html($.i18n("msg_parson_not_solution"))}),10),e.lines[0].distractHelptext&&(e.view.setAttribute("data-toggle","tooltip"),e.view.setAttribute("title",e.lines[0].distractHelptext)),e.disable(),e.inSourceArea())$(e.view).css({"border-color":"#000","background-color":"#fff"}),$(e.view).animate({opacity:.3,"border-color":"#d3d3d3","background-color":"#efefef"},{duration:2e3,complete:function(){$(e.view).css({"border-color":"","background-color":""})}});else{var i=this.sourceArea.getBoundingClientRect(),s=e.pageXCenter()-1,n=e.pageYCenter(),r=i.left+window.pageXOffset+i.width/2,o=i.top+window.pageYOffset+e.view.getBoundingClientRect().height/2,a=e.slideUnderBlock();void 0!==a&&(o+=a.view.getBoundingClientRect().height+20,o+=parseInt($(a.view).css("top")));var l=this;$(e.view).css({"border-color":"#000","background-color":"#fff"}),$(e.view).animate({opacity:1},{duration:4*Math.sqrt(Math.pow(o-n,2)+Math.pow(r-s,2))+500,start:function(){l.moving=e,l.movingX=s,l.movingY=n,l.updateView()},progress:function(e,t,i){l.movingX=s*(1-t)+r*t,l.movingY=n*(1-t)+o*t,l.updateView()},complete:function(){delete l.moving,delete l.movingX,delete l.movingY,l.updateView(),$(e.view).animate({opacity:.3,"border-color":"#d3d3d3","background-color":"#efefef"},{duration:1e3,complete:function(){$(e.view).css({opacity:"","border-color":"","background-color":""}),$(e.view).addClass("disabled")}})}})}}removeIndentation(){var e=$(this.messageDiv);this.messageDiv.style.visibility="visible",e.attr("class","alert alert-info"),setTimeout((()=>{e.html($.i18n("msg_parson_provided_indent"))}),10);for(var t,i,s=200,n=0;n0&&t.indexOf(s)<0)return s;return null}getFurthestBlock(){for(var e=this.solutionBlocks(),t=this.answerBlocks(),i=0,s=0,n=null,r=null,o=0,a=null,l=0,h=0;h0&&(a=e[o-1],l=t.indexOf(a),(s=Math.abs(h-l))>i&&(i=s,n=r));return n}combineBlocks(){var e=this.solutionBlocks(),t=this.answerBlocks(),i=this.sourceBlocks(),s=$(this.messageDiv);this.messageDiv.style.visibility="visible",s.attr("class","alert alert-info"),setTimeout((()=>{s.html($.i18n("msg_parson_combined_blocks"))}),10);var n=null,r=null;null==(r=this.getSolutionBlockInSource())&&(r=this.getFurthestBlock());var o=e.indexOf(r);n=e[o-1];var a=t.indexOf(n),l=t.indexOf(r),h=!1;a>=0&&l>=0?h=a+1!==l:a<0&&l<0?h=(a=i.indexOf(n))+1!==(l=i.indexOf(r)):(h=!0,a<0&&(a=i.indexOf(n)),l<0&&(l=i.indexOf(r)));var c=l3?(alert($.i18n("msg_parson_will_combine")),this.combineBlocks(),this.logMove("combinedBlocks")):(alert($.i18n("msg_parson_three_blocks_left")),this.canHelp=!1):(alert($.i18n("msg_parson_remove_incorrect")),this.removeDistractor(e),this.logMove("removedDistractor-"+e.hash()))}helpMe(){this.clearFeedback(),this.numDistinct<3?alert($.i18n("msg_parson_atleast_three_attempts")):(this.gotHelp=!0,this.makeEasier())}dateFromTimestamp(e){var t=new Date(e);if(isNaN(t.getTime())){var i=e.split(/[- :]/);t=new Date(i[0],i[1]-1,i[2],i[3],i[4],i[5])}return t}shuffled(e){for(var t,i,s=e.length,n=e.slice();0!==s;)i=Math.floor(Math.random()*s),t=n[s-=1],n[s]=n[i],n[i]=t;return n}enterKeyboardMode(){$(this.keyboardTip).show(),$(this.sourceLabel).hide(),$(this.answerLabel).hide(),this.clearFeedback()}exitKeyboardMode(){$(this.keyboardTip).hide(),$(this.sourceLabel).show(),$(this.answerLabel).show()}clearFeedback(){$(this.answerArea).removeClass("incorrect correct");for(var 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A=Math.round(f/this.options.pixelsPerIndent);for(A<0?A=0:A>this.indent?A=this.indent:f=A*this.options.pixelsPerIndent,m=this.movingY-this.answerArea.getBoundingClientRect().top-window.pageYOffset,this.moving.indent=A,y=0;y=10&&(o+=" "),e[a].addLabel(o,0),s++}n+this.pairedBins.length>=10&&(this.areaWidth+=5,$(this.sourceArea).css({width:$(this.sourceArea).width()+5}),$(this.answerArea).css({width:$(this.answerArea).width()+5}))}resetView(){if(this.clearFeedback(),!0===this.options.scaffolding){const e=document.querySelector(`#copy-answer-button-${this.divid}`);e&&e.classList.add("copy-button-hide")}var e,t=document.body.scrollTop;for(let t=0;t{$.i18n().load({"pt-br":{msg_parson_check_me:"Verificar",msg_parson_reset:"Resetar",msg_parson_help:"Ajuda",msg_parson_too_short:"Seu programa é muito curto. 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  • Quizzes
    - +
  • Recitations
    diff --git a/course/course-documents.html b/course/course-documents.html index 5dbf87d..133bd5f 100755 --- a/course/course-documents.html +++ b/course/course-documents.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/frontmatter.html b/course/frontmatter.html index 7560115..b5d88ae 100755 --- a/course/frontmatter.html +++ b/course/frontmatter.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/index.html b/course/index.html index d4bb52f..7b1a005 100755 --- a/course/index.html +++ b/course/index.html @@ -2,7 +2,7 @@ - + diff --git a/course/lunr-pretext-search-index.js b/course/lunr-pretext-search-index.js index 6e56c4f..2420a1a 100755 --- a/course/lunr-pretext-search-index.js +++ b/course/lunr-pretext-search-index.js @@ -801,6 +801,42 @@ var ptx_lunr_docs = [ "title": "", "body": " A diagnostic test is developed to detect a disease present in 1.3% of the population. For a patient who has the disease, the test will accurately give a positive result 62% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.4% of the time. For a patient who receives a positive test, what is the probability they have the disease? Let be the event of receiving a positive test result and be the event of having the disease. The given information is: , and . Then, using Bayes' Theorem: " }, +{ + "id": "quiz-02", + "level": "1", + "url": "quiz-02.html", + "type": "Worksheet", + "number": "", + "title": "Quiz 2", + "body": " Quiz 2 The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app. Show all work unless instructed otherwise. Write either True or False for each of the following statements. No justification is required. Let be integers with . Then . True. Suppose is a continuous random variable with pdf and cdf . Then . True. Suppose is a random variable taking values between 0 and 6. Then . False. Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? Since , and are not independent. Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . " +}, +{ + "id": "quiz-02-3-1", + "level": "2", + "url": "quiz-02.html#quiz-02-3-1", + "type": "Worksheet Exercise", + "number": "1", + "title": "", + "body": " Write either True or False for each of the following statements. No justification is required. Let be integers with . Then . True. Suppose is a continuous random variable with pdf and cdf . Then . True. Suppose is a random variable taking values between 0 and 6. Then . False. " +}, +{ + "id": "quiz-02-3-2", + "level": "2", + "url": "quiz-02.html#quiz-02-3-2", + "type": "Worksheet Exercise", + "number": "2", + "title": "", + "body": " Consider the joint distribution for and below. Joint Distribution 0.1 0.05 0.1 0.3 0.15 0.3 Find the marginal distributions for and . Are and independent? Since , and are not independent. " +}, +{ + "id": "quiz-02-4-1", + "level": "2", + "url": "quiz-02.html#quiz-02-4-1", + "type": "Worksheet Exercise", + "number": "3", + "title": "", + "body": " Suppose a continuous random variable taking values in has cdf . Find the pdf . Find . " +}, { "id": "recitation-calculus-review", "level": "1", @@ -851,7 +887,7 @@ var ptx_lunr_docs = [ "level": "2", "url": "recitation-calculus-review.html#thm-FTC", "type": "Theorem", - "number": "90", + "number": "91", "title": "Fundamental Theorem of Calculus.", "body": " Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: " }, diff --git a/course/notes-01-13.html b/course/notes-01-13.html index 08a6348..1669089 100755 --- a/course/notes-01-13.html +++ b/course/notes-01-13.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/notes-01-15.html b/course/notes-01-15.html index 7b0f0c8..91f1d28 100755 --- a/course/notes-01-15.html +++ b/course/notes-01-15.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/notes-01-20.html b/course/notes-01-20.html index 3f15bcd..50efac3 100755 --- a/course/notes-01-20.html +++ b/course/notes-01-20.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/notes-01-22.html b/course/notes-01-22.html index 11e90a6..a34e5c4 100755 --- a/course/notes-01-22.html +++ b/course/notes-01-22.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/notes-01-27.html b/course/notes-01-27.html index 5cc4f25..a3b6b64 100755 --- a/course/notes-01-27.html +++ b/course/notes-01-27.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/notes-01-29.html b/course/notes-01-29.html index 0766358..b54c91d 100755 --- a/course/notes-01-29.html +++ b/course/notes-01-29.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    @@ -481,7 +484,7 @@ E(X) = 0 \cdot \underbrace{\Pr(X = 0)}_{1 - p} + 1 \cdot \underbrace{\Pr(X = 1)}
    \begin{gather*} -\E)N) = \sum_{i=0}^n i\cdot b(i; n, p) = \sum_{i = 0}^n i \cdot {n \choose i} p^i (1-p)^{n - i}. +\E(N) = \sum_{i=0}^n i\cdot b(i; n, p) = \sum_{i = 0}^n i \cdot {n \choose i} p^i (1-p)^{n - i}. \end{gather*}
    Gross.
    diff --git a/course/quiz-01-printable.html b/course/quiz-01-printable.html index d5ea4b7..2729a78 100755 --- a/course/quiz-01-printable.html +++ b/course/quiz-01-printable.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false; -PrevUpNext +PrevUpNext
  • Quizzes
    - +
  • Recitations
    @@ -333,7 +336,7 @@ B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \}
    🔗
    🔗
    -PrevTopNext +PrevTopNext
    diff --git a/course/quiz-01.html b/course/quiz-01.html index 1b5dcdc..e7dd4c3 100755 --- a/course/quiz-01.html +++ b/course/quiz-01.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
    -PrevUpNext +PrevUpNext
  • Quizzes
    - +
  • Recitations
    @@ -317,7 +320,7 @@ B \amp = \{ (0, 1), (0, 3), (0, 5), (1, 0), (3, 0), (5, 0) \}
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    🔗
    -PrevTopNext +PrevTopNext
    diff --git a/course/quiz-02-printable.html b/course/quiz-02-printable.html new file mode 100755 index 0000000..ffe1902 --- /dev/null +++ b/course/quiz-02-printable.html @@ -0,0 +1,385 @@ + + + + + + + + + + + + + + + + +Quiz 2 + + + + + + + + + + + + + + + + + + + + +Skip to main content
    +

    Introduction to Probability and Statistics for the Life Sciences Spring 2026 MATH 1044

    +

    Andy Eisenberg

    +
    +
    + +
    +
    +

    Print preview

    +
    +
    +
    +Paper size +
    + +
    + +
    +
    +

    +Worksheet Quiz 2
    +

    +
    The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.
    🔗
    +
    +
    Show all work unless instructed otherwise.
    🔗
    +

    1.

    +
    Write either True or False for each of the following statements. No justification is required.
    🔗
    +
    +

    (a)

    +
    Let \(n, k\) be integers with \(0 \leq k \leq n\text{.}\) Then \({n\choose k} = {n \choose n - k}\text{.}\)
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    +
    +
    Solution.
    +
    True.
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    +
    +
    🔗
    +
    +
    🔗

    (b)

    +
    Suppose \(X\) is a continuous random variable with pdf \(f(x)\) and cdf \(F(x)\text{.}\) Then \(\displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}\text{.}\)
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    +
    +
    Solution.
    +
    True.
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (c)

    +
    Suppose \(X\) is a random variable taking values between 0 and 6. Then \(\E(X) = 3\text{.}\)
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    +
    +
    Solution.
    +
    False.
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    +
    +
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    2.

    +
    +
    Consider the joint distribution for \(X\) and \(Y\) below.
    🔗
    +
    +
    Table 90. Joint Distribution
    🔗
    + + + + + + + + + + + + + + + + + + +
    \(X = 0\)\(X = 1\)\(X = 2\)
    \(Y = 0\)0.10.050.1
    \(Y = 1\)0.30.150.3
    +
    +

    (a)

    +
    Find the marginal distributions for \(X\) and \(Y\text{.}\)
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{align*} +\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\ +\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \\ +\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 \\ +\Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\ +\Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 +\end{align*} +
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (b)

    +
    Are \(X\) and \(Y\) independent?
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{align*} +\Pr(X = 2, Y = 0) \amp = 0.2 \\ +\Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1 +\end{align*} +
    +
    Since \(0.2 \neq 0.1\text{,}\) \(X\) and \(Y\) are not independent.
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗
    🔗

    3.

    +
    Suppose a continuous random variable \(X\) taking values in \([0, 1]\) has cdf \(F(x) = 2x^2 - x^4\text{.}\)
    🔗
    +
    +

    (a)

    +
    Find the pdf \(f(x)\text{.}\)
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{gather*} +f(x) = F'(x) = 4x - 4x^3. +\end{gather*} +
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (b)

    +
    Find \(\E(X)\text{.}\)
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    +
    +
    Solution.
    +
    +
    +\begin{align*} +\E(X) \amp = \int_0^1 x f(x)\ dx \\ +\amp = \int_0^1 x (4x - 4x^3)\ dx \\ +\amp = \int_0^1 4x^2 - 4x^4\ dx \\ +\amp = \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 \\ +\amp = \left(\frac{4}{3} - \frac{4}{5}\right) - (0) \\ +\amp = \frac{8}{15} \approx 0.533. +\end{align*} +
    +
    🔗
    +
    +
    🔗
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    +
    🔗
    🔗
    🔗
    +
    +
    +PrevTopNext +
    +
    +
    + +
    + + diff --git a/course/quiz-02.html b/course/quiz-02.html new file mode 100755 index 0000000..47022a2 --- /dev/null +++ b/course/quiz-02.html @@ -0,0 +1,369 @@ + + + + + + + + + + + + + + + + +Quiz 2 + + + + + + + + + + + + + + + + + + + + +Skip to main content
    +

    Introduction to Probability and Statistics for the Life Sciences Spring 2026 MATH 1044

    +

    Andy Eisenberg

    +
    +
    + +

    +Worksheet Quiz 2
    +

    +
    The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.
    🔗
    +
    +
    Show all work unless instructed otherwise.
    🔗
    +

    1.

    +
    Write either True or False for each of the following statements. No justification is required.
    🔗
    +
    +

    (a)

    +
    Let \(n, k\) be integers with \(0 \leq k \leq n\text{.}\) Then \({n\choose k} = {n \choose n - k}\text{.}\)
    🔗
    +
    +
    Solution.
    +
    True.
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (b)

    +
    Suppose \(X\) is a continuous random variable with pdf \(f(x)\) and cdf \(F(x)\text{.}\) Then \(\displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}\text{.}\)
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    +
    +
    Solution.
    +
    True.
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (c)

    +
    Suppose \(X\) is a random variable taking values between 0 and 6. Then \(\E(X) = 3\text{.}\)
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    +
    +
    Solution.
    +
    False.
    🔗
    +
    +
    🔗
    +
    +
    🔗
    🔗

    2.

    +
    +
    Consider the joint distribution for \(X\) and \(Y\) below.
    🔗
    +
    +
    Table 90. Joint Distribution
    🔗
    + + + + + + + + + + + + + + + + + + +
    \(X = 0\)\(X = 1\)\(X = 2\)
    \(Y = 0\)0.10.050.1
    \(Y = 1\)0.30.150.3
    +
    +

    (a)

    +
    Find the marginal distributions for \(X\) and \(Y\text{.}\)
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{align*} +\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\ +\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \\ +\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 \\ +\Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\ +\Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 +\end{align*} +
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (b)

    +
    Are \(X\) and \(Y\) independent?
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{align*} +\Pr(X = 2, Y = 0) \amp = 0.2 \\ +\Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1 +\end{align*} +
    +
    Since \(0.2 \neq 0.1\text{,}\) \(X\) and \(Y\) are not independent.
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗
    🔗

    3.

    +
    Suppose a continuous random variable \(X\) taking values in \([0, 1]\) has cdf \(F(x) = 2x^2 - x^4\text{.}\)
    🔗
    +
    +

    (a)

    +
    Find the pdf \(f(x)\text{.}\)
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{gather*} +f(x) = F'(x) = 4x - 4x^3. +\end{gather*} +
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗

    (b)

    +
    Find \(\E(X)\text{.}\)
    🔗
    +
    +
    Solution.
    +
    +
    +\begin{align*} +\E(X) \amp = \int_0^1 x f(x)\ dx \\ +\amp = \int_0^1 x (4x - 4x^3)\ dx \\ +\amp = \int_0^1 4x^2 - 4x^4\ dx \\ +\amp = \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 \\ +\amp = \left(\frac{4}{3} - \frac{4}{5}\right) - (0) \\ +\amp = \frac{8}{15} \approx 0.533. +\end{align*} +
    +
    🔗
    +
    +
    🔗
    +
    +
    🔗
    🔗
    🔗
    +
    +PrevTopNext +
    +
    +
    + +
    + + diff --git a/course/quizzes.html b/course/quizzes.html index 579414f..cfc0c66 100755 --- a/course/quizzes.html +++ b/course/quizzes.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    @@ -229,7 +232,10 @@ eBookConfig.enable_chatcodes = false;

    Chapter Quizzes

    -
    🔗
    +
    🔗
    PrevTopNext
    diff --git a/course/recitation-calculus-review-printable.html b/course/recitation-calculus-review-printable.html index b280cc0..971c495 100755 --- a/course/recitation-calculus-review-printable.html +++ b/course/recitation-calculus-review-printable.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    @@ -292,7 +295,7 @@ eBookConfig.enable_chatcodes = false;
    🔗
    🔗
    If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
    🔗
    +
    🔗
    If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
    🔗

    5.

    Practice with the following integrals:
    🔗
    diff --git a/course/recitation-calculus-review.html b/course/recitation-calculus-review.html index f63094e..1bdb224 100755 --- a/course/recitation-calculus-review.html +++ b/course/recitation-calculus-review.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    @@ -277,7 +280,7 @@ eBookConfig.enable_chatcodes = false;
    🔗
    🔗
    If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
    🔗
    +
    🔗
    If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
    🔗

    5.

    Practice with the following integrals:
    🔗
    diff --git a/course/recitations.html b/course/recitations.html index f874761..a1888d4 100755 --- a/course/recitations.html +++ b/course/recitations.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
    -PrevUpNext +PrevUpNext
  • Quizzes
    - +
  • Recitations
    @@ -231,7 +234,7 @@ eBookConfig.enable_chatcodes = false;
    🔗
    -PrevTopNext +PrevTopNext
    diff --git a/course/root-1-2.html b/course/root-1-2.html index 32ac1bb..20c872e 100755 --- a/course/root-1-2.html +++ b/course/root-1-2.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations
    diff --git a/course/syllabus.html b/course/syllabus.html index c075ecf..5319db1 100755 --- a/course/syllabus.html +++ b/course/syllabus.html @@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false; eBookConfig.enableScratchAC = false; eBookConfig.build_info = ""; eBookConfig.python3 = null; -eBookConfig.runestone_version = '7.11.11'; +eBookConfig.runestone_version = '7.11.12'; eBookConfig.jobehost = ''; eBookConfig.proxyuri_runs = ''; eBookConfig.proxyuri_files = ''; eBookConfig.enable_chatcodes = false; - + @@ -219,7 +219,10 @@ eBookConfig.enable_chatcodes = false;
  • Quizzes
    - +
  • Recitations