Page 197 - Probability and Statistical Inference

174 3. Multivariate Random Variables

from the Theorem 3.9.2 that for any fixed real number a, one has P{X ≤

3.9.2 Suppose that X has the Gamma(4, 1/2) distribution.
(i) Use Markov inequality to get an upper bound for P{X ≥ 12};
(ii) Use Bernstein-Chernoff inequality to get an upper bound for P{X ≥
12};
(iii) Use Tchebysheff’s inequality to obtain an upper bound for P{X ≥
12}.
3.9.3 (Exercise 3.6.6 Continued) Suppose that (X , X ) is distributed as
1
2
N (3, 1, 16, 25, 3/5).
2
(i) Use Tchebysheff’s inequality to get a lower bound for P{–2 < X <
2
10 | X = 7};
1
(ii) Use Cauchy-Schwarz inequality to obtain an upper bound for (a)
E[|X X |] and (b)
1 2
{Hint: In part (i), work with the conditional distribution of X | X = 7 and
2
1
from X , subtract the right conditional mean so that the Tchebysheff’s in-
2
equality can be applied. In part (ii), apply the Cauchy-Schwarz inequality in a
straightforward manner.}
3.9.4 Suppose that X is a positive integer valued random variable and
denote p = P(X = k), k = 1, 2, ... . Assume that the sequence {p ; k ≥ 1}
k
k
–2
is non-increasing. Show that p ≥ 2k E[X] for any k = 1, 2, ... . {Hint:
k
E[X] ≥ Refer to (1.6.11).}
3.9.5 Suppose that X is a discrete random variable such that P(X = –a) =
P(X = a) = 1/8 and P(X = 0) = 3/4 where a(> 0) is some fixed number.
Evaluate µ, σ and P(|X| ≥ 2σ). Also, obtain Tchebysheff’s upper bound for
P(|X| ≥ 2σ). Give comments.
3.9.6 Verify the convexity or concavity property of the following func-
tions.
(i) h(x) = x for –1 < x < 0;
5
3
(ii) h(x) = |x| for x ∈ ℜ;
(iii) h(x) = x e for x ∈ ℜ ;
4 –2x
+
(iv) h(x) = e –1/2x2 for x ∈ ℜ;
–1
+
(v) h(x) = x for x ∈ ℜ .
3.9.7 Suppose that 0 < α, β < 1 are two arbitrary numbers. Show that
{Hint: Is log(x) with x > 0 convex or concave?}