Solutions to Midterm
1. [20 marks] Suppose that X1 , , Xn are independent continuous random variables with
density
f (x; ) = x1
for 0 x 1
where > 0.
(a) The log-likelihood function is
n
ln L() = n ln() + ( 1)
ln(xi )
i=1
and its derivatives are
n
n
d
ln(x

Some final exam practice problems
Note: Some of these questions involve more extensive reflection than others and would not
therefore be appropriate for a two hour exam.
1. Suppose that X1 , X2 , are independent random variables with common density functi

Solutions Assignment #4 STA355H1S
1. (a) 1 F (x) = (1 + x ) ; multiplying by x (x ) = 1, we have
1 F (x) = x (1 + x ) x )
= x (1 + x ) .
If we dene h(t) = t then for close to 0, we have (by a Taylor series expansion)
h(1 + ) = h(1) + h (1) + o()
= 1 + o()

Solutions Assignment #3 STA355H1S
1. (a) Taking the scale parameter of the Gamma distribution equal to 1, the mean of the
distribution is and the mode is 1 (for 1). The median can be computed in R using
the qgamma function. The comparison between the true

Solutions to Assignment #2 STA355H1S
1. (a) For a sample with 2m 1 observations, the median will be the m-th smallest in the
sample. If Xi X(m) then i = X(m+1) and for Xi X(m+1) , we have i = X(m) .
(b) From part (a),
1 n
1
i = (X(m) + X(m+1) )
=
n i=1
2

Solutions to Assignment #1 STA355H1S
1. (a) By the Delta Method
n(g(Xn ) g() N (0, [g ()]2 2 ()
d
and so [g ()]2 2 () = 1, which means that
g () =
1
.
()
(b) For the Poisson distribution 2 () = and so
1
g () = .
We can take g() = 2 (or more generally g()

Solutions to Midterm Examination STA355H1S
1. (a) The log-likelihood function is
n
ln L() = n ln()
xi
i=1
and its rst two derivatives are
n
n
d
ln L() =
xi
d
i=1
n
d2
ln L() = 2
2
d
Thus the MLE is = 1/X and its estimated standard error is
d2
se() = 2 l

Midterm Examination STA355H1S
February 26, 2014
Instructions: Do all four questions in the examination booklets and show all your work.
Good luck!
1. [18 marks] Suppose that X1 , , Xn are independent exponential random variables with
density function
f (x

Midterm Examination STA355H1S
February 27, 2013
Instructions: Do all four questions in the examination booklets and show all your work.
Good luck!
1. [20 marks] Suppose that X1 , , Xn are independent continuous random variables with
density
f (x; ) = x1
f

Solutions: Final exam practice questions
1. (a) Note that
F (x) =
Z x
t1 dt = 1 x
1
for x 1. Then
P (Vn x) = F (x)n = (1 x )n
and the density of Vn is given by
d
P (Vn x) = n(1 x )n1 x1
dx
for x 1.
(b) The distribution function of n1/ Vn is given by
P (n1