SS2857 Probability and Statistics I | Reliable Papers

SS2857 Probability and Statistics IFall 2020Assignment #5SubmissionAll times are stated according to the time in London, Ontario. You are responsible for checkingthese times in your current location.Solutions must be submitted to Gradescope no later than 11:59pm on Friday December4.If you were enrolled in the course prior to September 15 then you have been automaticallyadded to Gradescope site and should have received notification at your official UWO e-mail.If you were not enrolled at this time or did not receive the e-mail then you can access theGradescope site with the entry code 9NWP2K.You are responsible for ensuring that your solutions are legible and that you associate yoursolutions with the correct questions. Submissions will not be graded if the TAs cannot readyour solutions or the solutions aren’t matched to the questions.Late submissions:ˆ Late assignments without illness self-reports or other accommodation will be subject to alate penalty 20 % per dayˆ Late assignments with illness self-reports or other accommodation should be submittedwithin 24 hours of the end of the accommodation period. Assignments submitted anylater will be subject to a late penalty of 20% per day.ˆ Assignments that are submitted after 11:59pm on Wednesday December 9 will not begraded, even with accommodation.ˆ Solutions will be available on OWL at 12:01am on Thursday December 10.StyleWritten AnswersQuestions requiring a written solution should be answered in full sentences and paragraphs.Solutions will not be graded for grammar and spelling, but your answer must be clear in orderto receive marks.Calculation Based QuestionsPlease follow these guidelines for all questions that involve calculations:ˆ Clearly define any variables that you use in your solutions that are not defined in the question itself. E.g., if you are working with heights of ten people and label them 1, . . . , 101then you might start the question by saying: “Let  denote the height of the th personin the sample.”.ˆ Show your work. You do not have to show every single step, but the logic should be clear.There should be no jumps in the calculations.ˆ End your solution with a sentence or two that clearly answers the question.Here is an example. Suppose that the question states: “Show that the 5th number in theFibonacci sequence is 5.” Here is my solution.Let ƒ denote the th number in the Fibonacci sequence. The sequence is definedsuch that ƒ = ƒ-1 + ƒ-2 for  > 2 with ƒ1 = 1 and ƒ2 = 1. Then:ƒ3 = ƒ2 + ƒ1= 1 + 1= 2ƒ4 = ƒ3 + ƒ2= 1 + 2= 3ƒ5 = ƒ4 + ƒ3= 3 + 2= 5.Therefore, the 5th number in the Fibonacci sequence is 5.Questions (70 points in total)This assignment covers the material in Section 4.2 – 4.7, 5.1–5.3, and 6.1 in your textbook/Recorded Lectures 24A/B to 32.1. The 1995 article Reliability of Domestic-Waste Biofilm Reactors suggested that the distribution of substrate concentration of influent to a reactor in mg/cm3, denoted by X, isnormally distributed with mean μ = .30 and standard deviation σ = .06.a) (5 points) Compute p = P(X ≤ .20) and then use this value to compute each of thefollowing probabilities:i. X > .20ii. X ≥ .20iii. .20 < X < .40.iv. (X – .30)2 > .01b) (5 points) Let Z be a standard normal random variable. Use a transformation of Z toshow that the pdf of X isƒ() =1.06p2π exp ‚- (.-0072 .30)2 Œ ,  2 ℜ.c) (5 points) Suppose that the data contained the 100 observations shown in the following normal probability plot. The solid line connects the points at the first and thirdquantiles of the data. The dashed line on the plot has the equation y = .06 + .30.20.10.20.30.40.5-2 -1 0 1 2Theoretical Percentile of a Standard NormalSample PercentileDoes this plot support the researchers’ claims about the distribution of X? Explainyour answer.2. The 2016 paper Modeling Travel Time for Reliability Analysis in a Freeway Network byN. Wu and J. Geistefeldt suggests that highway travel times can be modelled using ashifted gamma distribution. That is, the time for travelling a specific piece of highwayis modelled as T = X +  where the random variable X follows a gamma distribution withsome parameters α and β and  is some constant. As an example they consider thedistribution with  = 20, E(T) = 25, and V(T) = 16.(a) (5 points) Find the parameters α and β for this specific example.(b) (5 points) Derive the probability density function of T for this specific example.(c) (5 points) Derive the moment generating function of T for this specific example.(d) (5 points) Use the moment generating function to confirm the mean and variance ofT.3. Suppose that X and Y have joint probability density functionƒ(, y) =3 2(2 + y2), 0 <  < 1 and 0 < y < 1.a) (5 points) Show that the marginal pdfs of X and Y areƒX() =1 232 + 1 , 0 <  < 1andƒY(y) =1 23y2 + 1 , 0 < y < 1.b) (4 points) Find the conditional pdf of X given Y = y.c) (4 points) Show that the mean and variance of X and Y areE(X) = E(Y) = .625V(X) = V(Y) = .076.d) (4 points) Compute the correlation between X and Y.3e) (3 points) Are X and y independent? Provide three different justifications for youranswer based on the previous parts of this question.4. Suppose that X1, . . . , Xn form a random sample of exponential random variables all withrate λ.a) (5 points) Explain what the sampling distribution of X¯ = Pn =1 /n represents.b) (5 points) Note that the sum of independent and identically distributed exponentialrandom variables has a gamma distribution. Specifically, if S = Pn =1 X then S followsa gamma distribution with parameters α = n and β = 1/λ. Use this fact to find thesampling distribution of X¯.c) (5 points) Describe how the skewness of the sampling distribution changes as n increases. Does the result surprise you? Justify your answer.4

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