1. Use the least-square method to find a plane z  ax  by  c fitting the following
data:
zi
xi
yi
8
3
5
4
0
3
4
5
2
1
1
0
2. Tom‘s and John’s records in mathematics tests are given as follows.
TOM 63 67 65 56 60 55
JOHN 58 69 62 60 58 58
Who is the better one? Or their records are almost in the same?
3. The following data give the numbers of units of work done per day by 3 workmen
using two machines. Find (a) 95% confidence interval of machine-type effect and (b)
90% confidence interval of workman-skill effect. ( P ( z  1. 64)  0. 9 ,
P ( z  1. 96)  0. 95 for the normal distribution)
I
II
1 35 37
2 36 40
3 48 30
4. Given ( x1 , y1 ) , ..., ( x5 , y5 ) for curve fitting, do you think that the third-degree
polynomial y  ax3  bx 2  cx  d is superior to the second-degree one
y  ax2  bx  c ? How can you justify your answer? (Hint: Use F-distribution)
5. What are the differences between the sign test and the rank-sum test?
_
_
6. Show that y  y  r 
sy
sx
_
_
 ( x  x)  y  [
 ( x  x) y
i
i
i
_
 ( x  x)
_
]  ( x  x ) is the equation of the
2
i
i
least-square line of fitting ( xi , yi ) , i .
7. Is the Poisson density or the geometric density to fit the following data? The total
number of sampling are 100 and   0. 05. If both the models fit them, which is the
better one?
xi
0
yi
61 25 10 4
1
2
3
8. Give two exponential densities f ( x      exp( x) , x  0 , with   1 and 2,
respectively. Find out the Bayes solution to the multiple decision problem if
(1 )  (2 )  0.5, and a single observation is to be taken.
9. Construct a sequential test for H0 :   1 against H1 :   2 for a normal
variable with   0 . (Suppose   0.1 and   0. 2 )
10. Let X  29.8 for a sample of 50 from a normal density with   5 , and test H0 :
  30 against H1 :   29 with   0. 05.
11. For an exponential density f ( x      exp( x) , x  0 . How large should n
be chosen to guarantee to test H0 :   1 10 against H1 :   1 9 .
12. Find the estimators of p for the geometric density by the following methods:
(a) the maximum likelihood method, (b)the first moment method, (c)the second
moment method, and (d)Bayesian method for the density function  ( p)  1 ,
0  p  1.
13. Let X  10 for a random sample of size 25 from a normal density with 2  4 ,
and determine 90% confidence interval for  .
14. Let X  9.5 and
X
2
i
 2000 for a random sample of size 20 from a normal
density with   10 , and determine 90% confidence interval for 2 .
15. Let X be a random variable having a continuous probability density function
f ( x ) ,   X   . Suppose that the moment generating function of f ( x ) is
M x (t ) 



f ( x )etx dx 
et
, t  1. Calculate (a) EX , (b)VarX , (c)  x ( t )  E ( t x ) ,
2
1 t
and (d)  x ( t )  E ( eitx ) .
6 x  12 y
, 0  x  y  1, obtain
5
if z is defined by z  y x , and
16. Given the probability density function f ( x , y ) 
(a) f ( y x  , (b) E (Y x , (c) f ( z x  and E ( Z x 
(d) Cov ( X , Y ) .
17. Let X and Y be independent random variables having binomial densities with
parameters p  0. 2 , nx  5 and n y  10 , respectively. Let Z  X  Y , and
determine EZ and VarZ . Ans. EZ  3 , VarZ  2.4 .
18. Let X and Y be random variables having geometric densities with parameter
p  0. 3. Let Y  5 X , and determine EY and VarY .
19. Let X and Y be independent random variables having Poisson densities with
parameters x  2 and  y  3 , respectively. Let Z  X  Y , and determine (a) EZ
and VarZ , and (b) E ( X z ) and E (Y z ) .
6 x  12 y
, 0  x  y  1 . Let
5
W  Y  X , Z  X  Y , and determine (a)the density function f w, z ( w, z ) and the
20. Give the probability density function f ( x , y ) 
range of W and Z , (b) E (W z and E ( Z w .
21. There is a joint binormal density function
1
3
 exp[  ( 3x 2  4 xy  12 y 2  2 x  44 y  43)] ,
64
8 2
and then determine (a) ( x , y ) , (b) E ( X y  , (c) E (Y x , (d) Var ( X y , and
(e) Var (Y x  .
f ( x, y) 
22. Let X and Y be separately an exponential density and a Poisson’s distribution
with a common parameter   5. Determine EX , VarX , EY , and VarY .
23. Let X i have a Poisson density with the same   0. 3 , i  n . All the X i ‘s are
independent of each other. Define Sn  X1  X 2   X n , and then determine
P( Sn  10) for n  30 . (Hint: Apply the central limit theorem)
24. There was a test for AIDS with the property that 90% of those with AIDS reacted
positively whereas 5% of those without AIDS react positively. Assume that 1% of
patients in a hospital have AIDS. What is the probability that a patient selected at
random who reacts positively to this test actually has AIDS?
25. Let X and Y be independent random variables having the same uniform
density function on [0,1]. Determine the probability density function (p.d.f.) of Z if
Z
(a) X  1  exp(  ) , (b) Z  X  Y , and (c) Z  X  Y .

26. Let X 1 and X 2 be independent random variables with the same density function
f ( x )  e  x . Show that the random variables Y1  X1  X 2 , Y2 
X1
are also
X1  X 2
independent.
27. Let X have the uniform probability density function on [0,1] and Y have the
uniform probability density function on [0,x]. Determine the joint probability density
function of X and Y .
28. The speed of a molecule in a uniform gas at the equilibrium is a random variable
V whose probability density function is given by f ( v )  kv 2  exp( bv 2 ) , v  0 ,
where k is a constant and b depends on the temperature and the mass of the
molecule. Determine the probability distribution of the kinetic energy of the molecule
mv2
W
.
2
29. Consider a binary transmission system where, at the end of each symbol interval,
   n  Z , if "1" was sent
the output of the demodulator is: r  
, where α (>0) is a
   n  Z , if "0" was sent
constant, n is a Gaussian random variable with zero mean and variance σ2. Z is a
discrete random variable with probability distribution Pr(Z=z)=0.25, Pr(Z=0)=0.5,
Pr(Z=-z)=0.25. Assuming that symbols “1” and “0” are equally likely, derive an
expression for the probability of the bit error in terms of α, σ, and z.
Table:
Standard normal distribution:
1
P( z  )  0. 6292 , P ( z  1. 28)  0.8 , P ( z  1. 64)  0. 9 , and P ( z  1. 96)  0. 95.
3
 2  distribution:
  3, P(  2  7.815)  0. 05,
  4 , P(  2  9. 488)  0. 05,
  20, P(  2  10.851)  0. 05, and P(  2  31. 41)  0. 05.
Student‘s t  distribution:
  20, P ( t  1. 725)  0. 9 , and P ( t  2. 086)  0. 95 .