Question 1.
The time, T, from treatment to a recurrence for patients diagnosed with a disease is assumed to follow an exponential distribution with mean, ? = 875 days.
a) Determine the probability that a patient will make it through 600 days without a recurrence.
b) If a patient has been 600 days since treatment without a recurrence, what is the probability that the patient will make it through the next 600 days without a recurrence?
c) The data below give a summary of data on the recurrence times for 50 patients who were treated for the disease. The average recurrence time for these patients is t = 875. If the times are independent from patient to patient, and the data are assumed to follow the exponential distribution with mean ? = 875, give an expression for the probability of the data given in the table. (An expression only is fine here - do not compute an actual answer). Time to Recurrence (Days) (0,200] (200-600] (600-1200] (1200-2000] > 2000 Number of Patients 13 16 9 6 6
d) Compute expected frequencies for each of the cells in the table in part c). Do you feel that the exponential distribution provides an adequate description of these data?
e) Based only on the data in the table, and making assumptions only about the independence from patient to patient, estimate (i.e. give a good guess at a value for):
i) The probability that a patient will make it through 600 days without a recurrence.
ii) The probability that a patient who has been 600 days since treatment without a recurrence will make it through the next 600 days without a recurrence.
Question 2:
A point starts from the origin and on any move is equally likely to go one unit up, down. left or right. independently of previous moves. Let Xu, X2, X3, X, be random variables giving the number of moves up. down, left, and right respectively in a sequence of n moves.
a) In a sequence of it = S moves, what is the probability X1 = 2.X2 = 3,X3 = 1,X, = 2?
b) If n = S and X1 2.X2 = 3,X1 = 1, X4 = 2, what is the location of the point after the 8 moves?
c} Give an expression for the probability that the point is at the origin after a sequence of 8 moves. (Again, no need to evaluate).
d After yr moves, the point will have travelled a distance of n units "on the ground", but we want the "as the crow flies" distance. Let D be a random variable giving the Euclidean distance of the point from the origin. So if the coordinates of the point are (z,p) after n moves. then D =vi(x2 +g2). Give an expression for L.P in terms of XI, X2,X2, X4 and show that E(D2)= n.
Question 3:
Consider the Cereal 114% problem discussed in Module 07, Lecture 13. Let X0, Xt, X2, X3, ... be discrete random variable) that represent the total number of distinct prizes that have been seen up to and including box 0.1,2.3...., respectively. Define states in a Markov Chain to be {0.1,2.3.4.5.6}. representing the number of distinct prizes seen.
a) Let Pi, be the (i,j)tli clement of the transition matrix, P, tot the Markov Chain for the Cereal Box Problem. Note that i = 0.1 2 6 and j = 0.1 2 6 Give the matrix P.
b) Compute F (ding software - see tutorial notes from Welt I for methods), and explain clearly what the entries represent. Verify by hand the element PI.
e) Compute Pu) (really using aiaftivare - tee tutorial notes from Week I for methods). You started with 0 distinct prizes, what is the probability distribution for X10, the number of distinct prizes atter opening 10 boxes? Use this distribution to find E(X10).
d) If E..„ is the expected number of distinct prizes after the nth box is opened, 1 claim that E„ -4 5 4_46 + 1. Check me out using P4 for n 2, 3,4.
e) Using the claim in d). solve the recursion to get an expression for E,, and verify your result by comparing it to E10 calculated in c).
Question 4:
A random variable, X, with probability density function
f(x) = k/1+(x-θ)2 for -∞< x < ∞
has a Cauchy distribution with parameter 9. (It can be shown that the mean, and higher order moments of the Cauchy distribution do not exist - Math 648 material)!
a) Determine the value of k for which ,f(z) is a probability distribution. (H f triibradt a tan- I (z - + c).)
b) Sketch the Cauchy probability density function. sod show that that f(z) is symmetric about x 0.
c) Find the cumulative distribution function. F(x), for the Cauchy distribution. Explain how you would simulate an observation from the Cauchy distribution using an observation on a liniform(0,0 random variable.
d) Does a N(0,972) distribution "fit" the Cauchy? Let 0 = 0, Find the point, t, on the Cauchy distribution such that P(X < t) = OS and find the value of er such that if N(0.&) then P(Y < 1) 0.8.
e) Let s be another point on the Cauchy distribution with 61= 0. such that P(X < a) = 0.05. If Y N(0.o2), find P(Y < s) using your a from e). Hence show that the value of a found in e) no longer gives a good "fit" to the Cauchy. (The Cauchy has fatter tails than the Normal).
Question 5:
For any permutation of the integers 1,2,... it , let t, represent the value of the eth term in the permutation.
Define an indicator variable, Xt to be 1 if ti < ti-1 and 0 otherwise. for z = 2.3.....n.
a) What must the probability distribution for X1 be if 5 = X1 + X2 + is to represent the number of ascents in the sequence?
b) Assuming a random permutation of the integers from Ito n, find P(..Vi = I), for x = 2,3.........n.
c) Find E(Sn) and hence find the expected number of ascents in a random permutation of the Integers from 1 to n.
d) In a sequence of p zeros and q ones, the ith term, ti, is called a Mange point if ti ≠ ti-1, for = 2,3,4... ,p+q. For example, the sequence 0, I, 1,0,0.1,0, t has p = q= 4, and fire change point. t2,t4, t6, t7, t8. Define an Indicator variable Xi with X1 - I if t, 4, and 0 otherwise, fort = 2,3.... .p+q. Use the method of Indicator variables to show that the expected number of change points in a random sequence of p 0's and q l's is 2pq/(p+q)•
Question 6:
Let T be a random variable giving the time to failure of fluorescent bulbs produced by a manufacturer, and assume T follows an exponential distribution with mean, 0 = 35000 hours.
a) Find the median failure time for a bulb produced by this manufacturer. (The median is a value, rn, zebu that. P(T≤ m) = 0.5).
b) I Install two of these bulbs in a double spotlight. Let Tmax, be the time to failure of the last bulb In the pair to burn out. Assuming that the failure times are independent, find the cumulative distribution function. fmax (t). for Tmax and use that to get fmax(t), the probability density function for Tmax.
c) Let Tmin be the failure time of the First bulb in the pair to burn out. Find Fmin(t) the cumulative probability function for Tmin, and use that to find fmin(t), the probability density function for Tmin.
d) Find E(Tmax) and E(Tmin).
Solve any three questions out of six questions.