2016-01-25: Statistics 1: Tutorial 14
1. Consider simple linear regression; there is one explanatory variable and
i N (0; 2 ) i.i.d. i = 1; : : : ; n
Yi = 0 + 1 xi + i
where x1 ; : : : ; xn are not all equal. Express this as a Gaussian linear model
Y = X
15-10-12 Statistics 1 Tutorial 2
1. Let X1 ; : : : ; Xn be i.i.d. N (; 2 ). Find the distribution of:
2 k=1 k
2. Let Z and Y be independent random variables with standard normal and 2 (n) distributions respectively. Find the density function
2015-10-26: Statistics 1: Tutorial 4
1. (Delta Method) Let (Xn )n1 be a sequence of random variables and let
Prove that: If
pn(X ) n!
N (0; 2)
and g is a function, continuously dierentiable with g 0 () > 0 then
pn (g(X )
! N 0;
2015-10-19: Statistics 1: Tutorial 3
1. Let X = (X ; : : : ; Xd ) be a vector of i.i.d standard Gaussian random variables. Let A be a d d
matrix with det(A) > 0 and a d-vector. Find the distribution of
Y = A(X + ):
2. Show that every multivariate Gaussi
2015-12-07: Statistics 1: Tutorial 9
1. Let X1 ; : : : ; Xn be a random sample from exponential distribution with unknown parameter . Find
the MLE of the quantity P(X1 > t) and show that the estimator obtained is asymptotically normal.
Compute its asympto
2015-12-14: Statistics 1: Tutorial 10
1. Let (X1 ; : : : ; Xn ) be a Exp() random sample, where is unknown.
(a) Using the fact that X1 + : : : + Xn
Recall that (n; 21 ) = 22n .
Gamma(n; ), nd the distribution of T = 2 Pni=1 Xi.
(b) Construct a symmetric
2015-12-21: Statistics 1: Tutorial 11
1. Suppose that the number of goals scored in football matches (in a football league where defence
seems to be a priority) may be considered as observations on independent identically distributed
random variables. We
2015-11-23: Statistics 1: Tutorial 7
1. Let X1 ; : : : ; Xn be a sample from a Pareto(a; ) distribution. That is, the density is:
p(x; a; ) =
where a >
1( x > a ) ;
Show that the maximum likelihood estimators of the parameter
2015-11-09: Statistics 1: Tutorial 5
1. Let X1 ; : : : ; Xn be a random sample from the Gamma(; ) distribution, where ; > 0 are unknown
parameters. Set up the equations that the ML (maximum likelihood) estimators of and should
bML (the maximum likelihood
2015-11-16: Statistics 1: Tutorial 6
1. A medical test can be performed to indicate whether or not a person has Green Monkey Disease.
We know that the test gives false positive results with probability 0:01 and false negative results
with probability 0:01
2015-11-30: Statistics 1: Tutorial 8
1. Let Vn 2n . Show that L( Vn
pn) n!1 N (0;
(L denotes law).
2. Suppose that X1 ; : : : ; Xn are i.i.d. variables each with probability function
pX (0) = 2
pX (1) = 2(1 )
pX (2) = (1 )2
(a) Find a and b
15-10-05 Statistics 1 Tutorial 1
1. Let Y1 ; : : : ; Yn be an i.i.d. random sample with cumulative distribution function F . Dene the
empirical cumulative distribution function by:
Fn (x) =
1( 1;x] (Yk ):
(a) What is the distribution of nFbn
2016-01-18: Statistics 1: Tutorial 13
1. Suppose that X1 ; : : : ; Xn and Y1 ; : : : ; Yn are two random samples from Exp() and Exp() respectively.
Gamma(; ), W Gamma(; ), V ? W . Show that V V W Beta(; ).
Find the LRT of H : = versus H : =
(a) Let V
2016-01-11: Statistics 1: Tutorial 12
1. (Karlin - Rubin theorem) Let X p(:; ) where p(:; ) is a density. Assume there is a statistic T (x)
such that for all b > a , the likelihood ratio pp(xx;ab ) is increasing as a function of the statistic T (x).