Page 184 - Probability and Statistical Inference

3. Multivariate Random Variables 161

(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 E[X X ], , and
1 2
3.2.10 Suppose that X = (X , ..., X ) has the Mult (n, p , ..., p ) with 0 <
5
1
1
5
5
p < 1, i = 1, ..., 5, . What is the distribution of Y = (Y , Y , Y )
2
1
i
3
where Y = X + X , Y = X , Y = X + X ? {Hint: Use the Theorem 3.2.2.}
1 1 2 2 3 3 4 5
3.2.11 Suppose that X = (X , X , X ) has the Mult (n, p , p , p ) with 0 <
1
3
3
1
2
3
2
p < 1, i = 1, 2, 3, . Show that the conditional distribution of X 1
i
given that X + X = r is Bin (r, p /(p + p )). {Hint: Use the Theorem 3.2.3.}
1 2 1 1 2
3.2.12 Suppose that X and X have the joint pmf
1 2
with 0 < p < 1, q = 1 – p. Find the marginal distributions of X and X and
2
1
evaluate P{X – X ≤ 1}.
2 1
3.2.13 Suppose that X and X have the joint pmf
1 2
(i) Show that f (x ) = 1/21 (2x + 3) for x = 1, 2, 3;
1 1 1 1
(ii) Show that f (x ) = 1/7 (x + 2) for x = 1, 2;
2 2 2 2
(iii) Evaluate P{X ≤ 2} and P{X < 2}.
1 2
3.2.14 Suppose that X and X have the joint pdf given by
1 2

The surface represented by this pdf is given in the Figure 3.10.1. Show
that P{X X > a} = 1 – a + alog(a) for any 0 < a < 1. {Hint: Note that
1 2