Determine the probability that a patient will make it

Question

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, B = 875 days.

a) Determine the probability that a patient will make it through 600 days without a recurrence.

b) If a patent 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 Owl 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 875. If the times are independent from patient to patient, and the data are assumed to follow the exponential distribution with mean B = 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)	(200-6001	(600.12001	(120040001	> 2000
Number of Patients	13	16	0	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?

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 600days 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 X_u, 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 X_I, X2,X2, X4 and show that E(D²)= n.

Question 3)

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 tr_iibradt 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 4)

Let 7' 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 _S m) = 0.5).

b) I Install two of these bulbs in a double spotlight. Let T_max, 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. bmax0. for T_max and use that to get f_max(t), the probability density function for T_max.

c)  Let 70 be the failure time of the First bulb in the pair to burn out. Find & At), the cumulative probability function for T_max, and use that to find f_min(t), the probability density function for T_min.

d)  Find E(T_max.) and E(T_min,).