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, θ = 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 θ = 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).

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 X₁, X₂, X₃, X₄ 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 n = 8 moves, what is the probability X₁ = 2, X₂ = 3, X₃ = 1, X₄ = 2?

b) If it = 8 and X₁ = 2, X₂ = 3, X₃ = 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 It moves, the point will have travelled a distance of it 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 (x, y) after n moves, then D = s² + y²). Give an expression for D² in terms of X₁, X₂, X₃, X₄ and show that E(D²) = n.