Math230, 2017-18: Workshop, Coursework & Quiz Questions Coursework and Quiz marks The maximum mark obtainable through the handwritten coursework is 15 for CW01CW09. Each week there are two or three standard coursework questions with marks summing to 15. Each week there is also a challenge question with 5 marks available. Each week you can obtain the 15 marks by answering the standard questions perfectly; however if you drop a few marks on these, then you might regain them through the challenge question. T = min (15, S + C) . where T =total marks, S =standard marks and C =challenge marks. There is also a Revision Quiz that revises key material from the first year module on probability, and further quizzes associated with CW03-CW09. You should answer the revision quiz on Moodle by 11:59pm on the Sunday at the end of Week 2; it is worth a total of 30 marks. For QZ03-QZ09, you should answer QZ0k by 11:59pm on the Sunday at the end of Week k; each such quiz is worth 5 marks. 1 MATH230 Week 01 - Moodle-assessed problems QZ k is due 11:59pm on the Sunday of Week k, except for the revision quiz (k=1), which is due on the Sunday ending Week 2. Numerical answers should be entered as decimals and accurate to 3dp. Revision Quiz M01.1 Basic probability Nigel has 3 cards; the first has a triangle on one side, the second has a square and the third has a circle. The other side of each card is blank. Showing you only the blank sides, Nigel asks you to choose a card at random. If you choose the triangle he rolls an unbiased four-sided die (i.e. with the numbers 1-4 on the faces, with each face equally likely to be chosen); if you choose the square he rolls a six-sided die and if you choose the circle he rolls an eight-sided die. The end result of the experiment is a shape and a number. However, Nigel does not tell you which shape you chose and does not let you see him roll the die, he says he will simply call out the final number after a dramatic pause. (i) What is the probability that you choose a triangle and the final number is a 3? (ii) What is the probability that the final number is a 3? (iii) What is the probability that the final number is not a 3? (iv) What is the probability that the shape is a triangle or the final number is 3 or both? (v) The final number is a 3 (Nigel has just called it out); what is the probability that your shape was a triangle? M01.2 Discrete random variable A discrete random variable, X, takes a value x {1, 2, 3, 4} with a probability of x/10. It cannot take any other value. (i) What is P (X 3)? (ii) What is E [X]? (iii) What is E [X 2 ]? (iv) What is E [1/X]? (v) What is Var [X]? 2 M01.3 Continuous random variable A continuous random variable, Y , has a cumulative distribution function of FY (y) = c 1/(1 + y)2 for y > 0, and 0 otherwise. (i) What is c? (ii) What is fY (3) (the density, evaluated at y = 3)? (iii) What is P (1 Y 3)? (iv) What is E [1 + Y ]? (v) What is the third quartile of Y ? M01.4 Variance, standard deviation and linearity of expectation variable, Z, has E [Z] = 4 and Var [Z] = 9. A random (i) What is E [10 2Z]? (ii) What is E [Z 2 ]? (iii) What is E [(10 2Z)2 ]? (iv) What is StdDev [Z]? (v) What is StdDev [10 2Z]? M01.5 Colds The number of times that an individual contracts a cold in a given year follows a Poisson distribution with the expected number per year being 5. A new drug has been introduced which changes the distribution to Poisson with expected number 3 per year for 75% of the population. For the other 25% of the population the drug has no appreciable effect on colds. Let N be the number of colds in a year and B be the event that the drug is beneficial for you. (i) The random variable that counts the number of colds when the drug is not beneficial is represented by (A) B | N , (B) N | B, (C) N | B C , (D) N, B C , (E) B | N . (ii) The distribution of the random variable in (i) is (A) Poisson(3), 3 (B) Poisson(0.25), (C) Poisson(0.75), (D) Poisson(5), (E) Poisson(1). (iii) You try the drug for a year and have 2 colds. What is the approximate probability that the drug is beneficial for you? (A) 0.75, (B) 0.224, (C) 0.051, (D) 0.889, (E) 1/2. M01.6 Modelling Which of the following distributions is the most appropriate for modelling each of these situations. (A) Poisson, (B) Bernoulli, (C) Continuous Uniform, (D) Discrete Uniform, (E) Exponential, (F) Binomial, (G) Geometric. (i) The number of people you need to ask before you find someone who prefers crisps to chocolate. (ii) The number of people in the queue for tickets, excluding you, when you first arrive at the cinema. (iii) The time from when you join the queue until the person at the very front of the queue has been served. (iv) The seat number (i.e. position from left to right, rather than row) of the oldest person in the cinema. (v) The time during the film showing at which your mobile phone goes off (you accidentally leave it on, and it does go off). (vi) The number of vegetarians in your Math230 tutorial group (excluding you). (vii) Whether there are any vegetarians in your Math230 tutorial group (excluding you). 4 MATH230 Week 01 - Workshop problems Please have a go at the problems before the workshop so that in the workshop you can focus on those problems which you had trouble with. W01.1 Jabberwocky The brave knight is lost in the Tulgey Wood. Three paths lead on from where he stands at the Tumtum tree, clutching his vorpal sword. Unbeknownst to him, all three paths lead out of the wood; however, just off the left path lies the Jubjub bird, sleeping. Two times in three a knight walking this path would pass the bird without waking it, but if awoken the Jubjub bird would carry him up to its nest of hungry young. Along the right path lurks the frumious Bandersnatch against which the knight has no hope of prevailing. The middle path leads to the Jabberwock; if the knight encounters the Jabberwock then he might fall victim to its jaws that bite, its claws that snatch or its beguiling eyes of flame; else he will defeat it with a snicker-snack of his vorpal sword; each of the four possibilities is equally likely. The knight is as likely as not to forge straight ahead down the centre path; the right and left paths appeal equally to him. (a) What is the probability that the knight makes it out of the tulgey wood alive? (b) The brave knight does not emerge from the wood; what is the probability that he encountered the Jabberwock? Hint: It may be helpful to define: 'A'='Alive', 'D'='Dead', L='left path', R='right' and C='centre'. W01.2 Pregnancy An ectopic pregnancy is twice as likely to develop when a pregnant woman is a smoker as it is when she is a non smoker. If 32% of women of childbearing age are smokers, what is the probability that a woman having an ectopic pregnancy is a smoker? You may find it helpful to define the following shorthand: A ectopic pregnancy, B smoker. W01.3 Discrete cdf A discrete random variable, R, has a cdf of 0 r<0 19 0r<1 fr (r) = 5/9 1r<2 2r (a) what is p (r> 1)? (b) What is P (R 1)? (c) What is the pmf of R? 5 (d) What is the median of R? W01.4 Discrete pmf to cdf pmf. Obtain the cdf of a random variable with the following \u001a pR (r) = (1 )r 0 r = 0, 1, 2, 3, . . . otherwise. Hint Study Figure 2.1 in your notes. Your function should work for any real number r R. First calculate F at the integers, being careful to distinguish the index of summation from the upper limit of summation. Secondly extend from the integers to the real line. W01.5 Continuous cdf Let the cdf of a continuous random variable X be x < 10 0 (x 10)/10 10 x 20 FX (x) = 1 x > 20 Find the pdf of X. W01.6 pdf The random variable X has pdf \u001a 2x 0 3/4 | X > 1/2). W01.7 pdfcst The random variable X has probability density function \u001a a(4 x2 ) 2 < x < 2 fX (x) = 0 otherwise. (i) Find the constant, a, and the cumulative distribution function of X and, hence or otherwise, evaluate P(1 < X < 0). (ii) Find the median, q5/32 and q27/32 of X. 6 MATH230 Week 01 - Assessed problems (coursework) Submission is due at 1pm on Tuesday in Week 2. Cdf, pmf, pdf and quantiles A01.1 Discrete pmfs to cdfs ables Obtain the cdf of the following discrete random vari- (i) R where for some parameter, (0, 1), P(R = 0) = 2 , P(R = 1) = (1 ), P(R = 2) = (1 ), P(R = 3) = (1 )2 , and for r / {0, 1, 2, 3}, P(R = r) = 0. (ii) R Unif(0, m). [4] Hint: The discrete uniform pmf is given near the start of Ch3 of your Math230 notes and in your first year notes. Study Figure 2.1 in your notes. Your function should work for any real number r R. First calculate F at the integers, being careful to distinguish the index of summation from the upper limit of summation. Secondly extend from the integers to the real line, either by stating ranges or using the int function, where relevant. A01.2 Butterfly The lifetime, X, in days of a species of butterfly has a density of \u001a fX (x) = 0 b/x5 x1 x > 1. (i) Find the cdf of X in terms of b, being careful to show your working. (ii) Write down the value of b and explain how you deduced this. (iii) Find the probability that (i) a butterfly lives for more than two days, (ii) a butterfly lives for more than half a day. (iv) Find the age which only 1% of butterflies reach. [7] 7 A01.3 Survivor of A discrete random variable, R, has a parameter (0, 1) and a pmf \u001a pR (r) = (1 )r 0 r = 0, 1, 2, 3, . . . otherwise. (i) Use the solution to WS01 Discrete pmf to cdf to write down the survivor function, SR (r), of R. (ii) Hence, or otherwise, find P(R > a + b|R > a) for integer a 0 and real number b 0. (iii) What is unusual about the formula for P(R > a + b|R > a)? [4] A01.4 Challenge A random variable X has the 0 1/3 + x/3 FX (x) = 1 following cdf. x<0 0x<1 x 1. by considering limi fx (x) (x 1 i),or otherwise, describe the random variable itself. [5] 8 math230 week 02 - workshop problems please have a go at before so that in you can focus on those which had trouble with. w02.1 discrete cdf moments 0 9 fr (r) =5/9 variable, r, has of r<0 0r<1 1r<2 2r usepmf calculated ws01 to answer following. (a) what is e [ra ] for any> 0? (b) What is Var [R]? (c) What is the skewness of R? W02.2 pdfcst moments Find the expectation, variance, and skewness for the random variable X in the question in WS1 pdfcst. W02.3 Poisson accidents. Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with expected number (i.e. parameter) 3. (a) Write the pmf of the Poisson distribution with this parameter value. (b) Find the probability that 3 or more accidents will occur today. (c) Using conditional probability, repeat part (b) if you know that at least one accident occurred today. W02.4 Indicator function. Let A be an interval on the real line. For a set A the function IA (x) of x is defined as \u001a 1 xA IA (x) = 0 x /A For the continuous random variable X, find E [IA (X)]. Hint: either consider the distribution of Y = IA (X) and use the known results for that or first consider an interval, A = [a, b], and draw the picture. 9 MATH230 Week 02 - Assessed problems (coursework) Submission is due at 1pm on Tuesday in Week 3. Cdf, pdf and moments A02.1 Skew2 A random variable, X, has an expectation of and a variance of 2 . (a) Show that its skewness can be written as \u0001 \u0002 3\u0003 1 3 3 E X . 3 (b) Hence derive an expression for E [X 3 ] in terms of and for a random variable whose pdf or pmf is symmetric about . [5] A02.2 Butterfly moments density of The lifetime, X, in days of a species of butterfly has a \u001a fX (x) = 0 b/x5 x1 x>1 (You discovered the value of b in CW01.) (a) Find the expectation of X a for a > 0. What constraints are there on a for the expectation to be finite? (b) Write down the expectation and variance of X. (c) Using the formula in Skew2, find the skewness of X. [6] A02.3 Unif or Exp Una and Ed have just phoned for a taxi. Una suggests modelling their waiting time as Unif(a, b) but Ed believes c + Exp() is better. Discuss the pros and cons of these options. Note: this question is not asking you to discuss how you would choose a, b or . [4] A02.4 Challenge Question For a sufficiently smooth function, f (x), Taylor expansion about some point, , gives 1 f (x) = f () + (x )f 0 () + (x )2 f 00 (), 2 10 for some (, x) between and x. Consider any function f with f 00 (x) \u0002 0\u0003for all x R and show that E [f (X)] f (E [X]). Hence relate E [X 2 ] to E [X] and E eX to E [X]. [5] 11