Page 294 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 271

5.2.3 Let X , ..., X be iid N(µ, 1), –∞ < µ < ∞. Consider the following
1
n
cases separately:

Is there some real number a in each case such that as n → ∞?
{Hint: Can one use the Theorems 5.2.4-5.2.5 together?}
5.2.4 Let X , ..., X be iid having the common pdf
1 n

Is there a real number a such that as n → ∞? {Hint: Is the Weak
WLLN applicable here?}
5.2.5 (Example 5.2.6 Continued) Suppose that X , ..., X are iid having the
n
1
following common distribution: P(X = i) = c/i ,i = 1, 2, 3, ... and 2 < p < 3.
p
1
Here, c ≡ c(p) (> 0) is such that . Is there a real number
a ≡ a(p) such that as n → ∞, for all fixed 2 < p < 3?
5.2.6 In the Theorem 5.2.4, part (i), construct a proof of the result that
Also, prove part (ii) when u = 0, v ≠ 0.
5.2.7 (Example 5.2.5 Continued) Let X , ..., X be iid Uniform (0, θ) with
1
n
θ > 0. Consider X , the largest order statistic. Find the range of γ(> 0) such
n:n
that as n → ∞. Does as n → ∞?
{Hint: In the first part, follow the approach used in the Example 5.2.10 and
later combine with the Theorem 5.2.4.}
5.2.8 Obtain the expression of using (5.2.23) when the X’s are iid
with
(i) Bernoulli(p), 0 < p < 1; (ii) Poisson(λ), λ > 0;
(iii) Geometric(p), 0 < p < 1; (iv) N(µ µ ), µ > 0.
2
In part (ii), show that V[S ] > V [ ]for all fixed λ(> 0).
2
5.2.9 (Example 4.4.12 Continued) Suppose that X , ..., X are iid random
1
n
variables having the negative exponential distribution with the common pdf
given by f(x) = σ exp{–(x – µ)/σ}I(x > µ) with –∞ < µ < ∞, 0 < σ < ∞. Let
–1
X be the smallest order statistic and T = Show that
n:1 n