Reference no: EM13671353

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.

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 3

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, ? = 35000 hours.

a) Find the median failure time for a bulb produced by this manufacturer. (The median is a value, m, such 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).

Question 4

A random variable, X, with probability density function f(x) = k 1 + (x ? ?) 2 , for ? ? < x < ? has a Cauchy distribution with parameter ?. (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(x) is a probability distribution.

b) Sketch the Cauchy probability density function, and show that that f(x) is symmetric about x = ?.

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 Uniform(0,1) random variable.

d) Does a N(?, ?2 ) distribution "fit" the Cauchy? Let ? = 0. Find the point, t, on the Cauchy distribution such that P(X ? t) = 0.8 and find the value of ? such that if Y ? N(0, ?2 ) then P(Y ? t) = 0.8.

e) Let s be another point on the Cauchy distribution with ? = 0, such that P(X ? s) = 0.95. If Y ? N(0, ?2 ) find P(Y ? s) using your ? from

f). Hence show that the value of ? found in e) no longer gives a good "fit" to the Cauchy. (The Cauchy has fatter tails than the Normal).