I am in need of an expert who can sit my exam for me at 3:55pm AEST time. Its an advanced statistics course and the exam is 1 hr 45mins long. I will provide course content and practice questions as attached files for your viewing.
Practice 1 Answers.pdf Practice 1 Questions.pdf Semester One Final Examination, 2018 STAT7003 Probability & Statistics 1. The Australian cricket team is implementing various “enhancement” (cheating) strategies to increase their chance of winning. The probability of being “caught out” in any one match is 0.02. (a) What is the probability that the team is not caught out at or before the 100th match? [3] (b) Suppose the team has not been caught out in the first 50 matches. What is the probability that they will be caught out at or before match 100? [3] Page 2 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 2. The number of visitors on any day to the Museum of Uninteresting Things is known to have a Poisson distribution with expectation 10. The admission charge is $3.00 per person, and it costs $16.00 per day to run the museum. (a) Determine the probability that the museum makes a profit on a day. [3] (b) Determine the expected profit. [2] Page 3 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 3. Let X be a random variable with a Weibull distribution with parameters α > 0 and λ > 0. This means that the cdf F of X is given by F (x) = { 1− e−(λx)α x ≥ 0 0 otherwise. (a) Formulate an algorithm to simulate from this distribution using the inverse-transform method, assuming that you have a method for drawing from the U(0, 1) distribution. [4] (b) Show that X can be written as X = Z1/α/λ, where Z ∼ Exp(1). [3] Page 4 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 4. We draw randomly a point in the unit circle in the following way: first draw an angle Θ uniformly in [0, 2π) and then draw independently a radius R uniformly in [0, 1]. Let X = R cos Θ and Y = R sin Θ be the corresponding Cartesian coordinates. (a) Give the joint pdf of R and Θ. [1] (b) Show that Jacobian (=determinant of the matrix of Jacobi) of the transformation( r θ ) 7→ ( x y ) = ( r cos θ r sin θ ) is r = √ x2 + y2. [2] Page 5 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics (c) Derive the joint pdf of X and Y . [3] Page 6 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 5. In a large population 60% of people own a mobile phone. An advertising company chooses at random (uniformly) 100 people from this population. Let X be the number of people in this group that do not own a mobile phone. (a) Explain why X can be modelled as having a Bin(100, 0.4) distribution. [2] (b) Approximate, using the Central Limit Theorem, the probability that more than half in the chosen group do not own a mobile phone. Express your answer in terms of the cdf, Φ, of the standard normal distribution. [4] Page 7 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 6. Albert rides his bike from home to university. His journey is comprised of three stages. The travel time for stage i is modelled as normally distributed with expectation µi and variance σ2i , i = 1, 2, 3, where (µ1, µ2, µ3) = (5, 12, 20) minutes, and (σ21, σ 2 2, σ 2 3) = (1, 9, 16). Bertha, Albert’s flat mate, prefers to take the bus to university. Her journey comprises two stages, the travel time (in minutes) for the first stage is modelled as N(15, 4) distributed, and for the second stage as N(20, 3). All travel stages are independent of each other. (a) What are the expected total travel times for Albert and Bertha? [2] (b) Find the probability that Albert arrives at university before Bertha, if they leave home at the same time. Express your answer in terms of the cdf, Φ, of the standard normal distribution. [5] Page 8 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 7. Consider the 3-state Markov chain described by the following transition graph. 1 2 3 1 1/3 2/3 1/2 1/2 (a) Give the 1-step transition matrix. [3] (b) Give the 2-step transition matrix. [3] Page 9 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics (c) Give the limiting distribution. [3] Page 10 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics 8. The pdf of the so-called shifted exponential distribution is as follows: f(x) = { λe−λ(x−α) x ≥ α 0 otherwise, where α ∈ R and λ > 0 are parameters. (a) If a random variable X has the above pdf, show that X can be written as X = α + Y , with Y ∼ Exp(λ). [3] Page 11 of 12 Semester One Final Examination, 2018 STAT7003 Probability & Statistics (b) LetX1, X2, . . . , Xn be a random sample from the above distribution. Use the Method of Moments to construct estimators for λ and α. [4] END OF EXAMINATION Page 12 of 12 frontcoverSTAT7003 exam18 Practice 2 Part 1.jpg Practice 2 Part 2.png Practice 2 Part 3.png