STAT 3083
Final Examination  December 1996  Time: 3 Hours 

Let T be a discrete random variable with its probability density function given by the following:
T  1  2  3  4  5 
f(t)  0.15  0.18  0.27  0.22  0.18 
(a)  Find E(T), Var(T). 
(b)  Find P[T > 3  T > 2] 
(c)  Find E(T  T > 2) 
(d)  Find the values of the cumulative distribution function:

(a)  what fraction of evenings will there be more requests than TV sets; 
(b)  if the owner charges $2.00 rental fee per set per evening, then what will be his expected rental revenue? 
(a)  Find c. 
(b)  Find the cumulative distribution function, F(y). 
(c)  Find P[Y < 1  Y < 1.5]. 
(d)  Find E(Y  Y < 1.5). 
(e)  Find the hazard rate function, , 0 < t < 2. 
(a)  Find the 'halflife' of such components, i.e. find 'b' such that P[ X > b] =1/2. 
(b)  What is the expected life of a 6 hours old component, E( X  X > 6) ? 
(c)  What is the probability that a 6 hours old component will survive another five hours, P[ X > 11  X > 6] ? 
(a)  What proportion of scores are over 100? 
(b)  Find the 57th percentile, i.e. the best score in the low 57% group. 


Find P[W < 2600].
(a)  Find P[W < 90]. 
(b)  Find P[W > 100]. 