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278 5. Concepts of Stochastic Convergence

that is distributed as Binomial(n, p), n ≥ 1. Apply the CLT to
show that

In other words, for practical problems, the Binomial(n, p) distribution can be
approximated by the N(np, np(1 – p)) distribution, for large n and fixed 0 < p
< 1.
5.3.19 (Exercise 5.3.18 Continued) Let X , ..., X be iid Bernoulli(p) with 0
n
< p < 1, n = 1. Let us denote 1 . Show that

{Hint: In view of the Exercise 5.3.18, would the Mann-Wald Theorem help?}
5.3.20 (Exercises 5.3.18-5.3.19 Continued) Suppose that X , ..., X are iid
1
n
Bernoulli(p), 0 < p < 1, n ≥ 1. Let us denote and W = V (1
n n
– V ). Show that
n

when p ≠ 1/2. {Hint: In view of the Exercise 5.3.18, would the Mann-Wald
Theorem help?}
5.3.21 Suppose that X , ..., X are iid with the common pdf
1 n 2
f (x) = cerp {3x – 4x } for x ∈ ℜ,
where c(> 0) is an appropriate constant.
(i) Find k such that as n → ∞;

(ii) Find a , b (> 0) such that as n → ∞.
n n
5.4.1 (Example 4.5.2 Continued) Suppose that the random variables X ,
i1
..., X are iid N(µ, σ , i, = 1, 2, and that the X ’s are independent of the X ’s.
2
1j
2j
inz
With n ≥ 2, n = (n , n ), let us denote
i 1 2
for i = 1, 2. Consider the two-sample t random variable t =
ν
which has the Student’s t distribu-

tion with ν ≡ ν(n) = (n + n –2) degrees of freedom. Does t converge to
1 2 ν