Page 183 - Probability and Statistical Inference

160 3. Multivariate Random Variables

(i) find the marginal pmf’s for X and X ;
1 2
(ii) find the conditional pmf’s f (x ) and f (x );
1/2 1 2/1 2
(iii) evaluate E[X | X = x ] and E[X | X = x ];
1 2 2 2 1 1
(iv) evaluate and
3.2.5 Prove the Theorem 3.2.2.
3.2.6 Prove the Theorem 3.2.3.
3.2.7 Consider an urn containing sixteen marbles of same size and weight
of which two are red, five are yellow, three are green, and six are blue. The
marbles are all mixed and then randomly one marble is picked from the urn
and its color is recorded. Then this marble is returned to the urn and again all
the marbles are mixed, followed by randomly picking a marble from the urn
and its color recorded. Again this marble is also returned to the urn and the
experiment continues in this fashion. This process of selecting marbles is
called sampling with replacement. After the experiment is run n times, sup-
pose that one looks at the number of red (X ), yellow (X ), green (X ) and blue
2
3
1
(X ) marbles which are selected.
4
(i) What is the joint distribution of (X , X , X , X )?
1 2 3 4
(ii) What is the mean and variance of X ?
3
(iii) If n = 10, what is the joint distribution of (X , X , X , X )?
1 2 3 4
(iv) If n = 15, what is the conditional distribution of (X , X , X ) given
1
2
3
that X = 5?
4
(v) If n = 10, calculate P{X = 1 ∩ X = 3 ∩ X = 6 ∩ X = 0}.
1 2 3 4
{Hint: In parts (i) and (v), try to use (3.2.8). In part (ii), can you use
(3.2.13)? For part (iii)-(iv), use the Theorem 3.2.2-3.2.3 respectively.}
3.2.8 The mgf of a random vector X = (X , ..., X ), denoted by M (t) with
1
X
k
t = (t , ..., t ), is defined as E[exp{t X + ... + t X }]. Suppose that X is
1
1
k
k
k
1
distributed as Mult (n, p , ..., p ). Then,
k 1 k
(i) show that M (t) = {p e } ;
t
n
X 1 k
(ii) find the mgf M (t) of U = X + X by substituting t = t = t and t
U 1 3 1 3 i
= 0 for all i ≠ 1, 3 in the part (i). What is the distribution of U?
(iii) use the procedure from part (ii) to find the mgf of the sum of any
subset of random variables from X.
3.2.9 Suppose that the random variables X and X have the following joint
2
1
pmf:
where x , x are integers, 0 ≤ x ≤ 3, 0 ≤ x ≤ 3, but 1 ≤ x + x ≤ 3. Then,
1 2 1 2 1 2