### STAT 3093 ----- FINAL EXAM ----- 1997 DEC. 10 ----- version 1

Time 3 Hours      Open Book - Calculator allowed      Attempt all questions

### Question ONE   (Short Items: Best 3 will be counted)   [24]

1. For normal distribution with mean = 60 and standard deviation = 6 ,  determine the probability content of the interval [62 to 67].
»»»» P{X<62} =0.6306;   P{X<67=}=0.8783;   P{X in 62 to 64}=0.2477

2. X and Y are independent random variables.
E{X}=100, Var{X}=15    E{Y}=100, Var{Y}=13

(a) Z = X-Y    Calculate the variance of Z.   »»»»  15+13=28
(b) Calculate Variance{Z+Y+Y+Y}   »»»» 15+4× 13

3. Suppose a sphere of diameter one is embedded in a 1 by 1 by 1 cube and its volume will be estimated by taking N random points in the cube and using the fact that the volume of the sphere equals the probability of hitting it.   For purposes of this question we know (though we usually don`t in such situations) the volume of the sphere,   pi/6=0.5235987755982...

(a) Out of N=90000 points, H hit the sphere. What is the std. dev. of the random variable (H/N)?
»»»» sqrt(0.5236×0.4764/90000) = 0.001664...

(b) What is the least N that will assure the standard deviation of (H/N) is no more than 10-5
»»»» Solve 10-5 = Z×sqrt(0.5236×0.4764/N)
»»»» N = (4×0.5264×1010) 5541939200 :: 5.54x109
((For grading this exam, Z=2 is accepted. For serious Monte Carlo work, Z should be at least 3.))

4. Here is a random sample from a large population.
35.0 35.2 35.3 35.3 40.0 40.6 50.0 60.0 60.0 70.0 70.5 85.0 95.0 95.3 95.8 140 200
Complete (use either table):
The population median is between ______ and _____ with probability ________.
»»»»  X[5]=40.0 to X[13]=95.0 with probability 0.9510   or
»»»»  X[3]=35.3 to X[15]=95.8 with probability 0.99765