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Here are some tips on how to set out your work, as well as how to complete the questions. None of the answers given below have been rounded to what the question states. 

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Question 1

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1a: $X\\sim Exp(\\var{Lambda1})$

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(i) $P(X= x)=\\lambda e^{-\\lambda x} \\rightarrow P(X= \\var{x1})=\\var{Lambda1}e^{-\\var{lambda1} (\\var{x1})} = \\var{Ans1} $

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(ii) $P(X= x)=\\lambda e^{-\\lambda x} \\rightarrow P(X= \\var{x2})=\\var{Lambda1}e^{-\\var{lambda1} (\\var{x2})} = \\var{Ans2} $

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(ii) $P(X= x)=\\lambda e^{-\\lambda x} \\rightarrow P(X= \\var{x3})=\\var{Lambda1}e^{-\\var{lambda1} (\\var{x3})} = \\var{Ans3} $

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1b: $X\\sim Exp(\\var{Lambda2})$

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(i) $P(X\\leq x)=1-e^{-\\lambda x} \\rightarrow P(X>\\var{x4})=1-P(X\\leq \\var{x4}) =1-(1-e^{-\\var{lambda2} (\\var{x4})}) = \\var{Ans4} $

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(ii) $P(X\\leq x)=1-e^{-\\lambda x} \\rightarrow P(X< \\var{x5})=P(X\\leq \\var{x5}-1)=P(X\\leq \\var{j1})=1-e^{-\\var{lambda2} (\\var{j1})} = \\var{Ans5} $

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(iii) $P(X\\leq x)=1-e^{-\\lambda x} \\rightarrow P(X\\geq \\var{x6})=1-P(X\\leq \\var{j2}) =1-(1-e^{-\\var{lambda2} (\\var{j2})}) = \\var{Ans6} $

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(iv) $P(X\\leq x)=1-e^{-\\lambda x} \\rightarrow P(X\\leq \\var{x7}) =1-e^{-\\var{lambda2} (\\var{x7})} = \\var{Ans7} $

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Question 2

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2a: $X\\sim Exp(\\var{Lambda3})$

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(i) $P(X\\leq x)=1-e^{-\\lambda x} \\rightarrow P(X>\\var{x8})=1-P(X\\leq \\var{x8}) =1-(1-e^{-\\var{lambda3} (\\var{x8})}) = \\var{Ans8} $

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(ii) $P(X\\leq x)=1-e^{-\\lambda x} \\rightarrow  P(X_1>\\var{x8}) \\times P(X_2>\\var{x8}) = (1-P(X_1\\leq \\var{x8}))\\times (1-P(X_2\\leq \\var{x8})) = (1-(1-e^{-\\var{lambda3} (\\var{x8})})) \\times (1-(1-e^{-\\var{lambda3} (\\var{x8})}))=\\var{ans9} $

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(a) Given that $X\\sim Exp(\\var{Lambda1})$. Find,

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(i) $P(X=\\var{x1})$

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Answer:[[0]]

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(ii) $P(X=\\var{x2})$

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Answer:[[1]]

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(iii) $P(X=\\var{x3})$

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Answer:[[2]]

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(b) Given that $X\\sim Exp(\\var{Lambda2})$. Find,

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(i) $P(X>\\var{x4})$

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Answer:[[3]]

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(ii) $P(X<\\var{x5})$

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Answer:[[4]]

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(iii) $P(X\\geq\\var{x6})$

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Answer:[[5]]

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(iv) $P(X\\leq\\var{x7})$

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Answer:[[6]]

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"suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{Ans3}", "maxValue": "{Ans3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{Ans4}", "maxValue": "{Ans4}", "correctAnswerFraction": false, 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not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{Ans6}", "maxValue": "{Ans6}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": 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(a) A group of students are gathering data for their modelling work. The group are measuring the time taken between arrivals of vechicles at a junction. The  junction has a parameter of {Lambda3}. 

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A pedestrian, who takes {x8} seconds to cross the road, sets off as one vechicle passes. Find,

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(i) the probability that they will complete the crossing before the next vechicle arrives.

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Answer:[[0]]

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(ii) the probability that they complete the crossing and return without a vechicle arriving. 

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Answer:[[1]]

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