Question 11.EX.32: Show that if X and Y have the same distribution then Var((X ......
Show that if X and Y have the same distribution then
Var((X + Y)/2) ≤ Var(X)
Hence, conclude that the use of antithetic variables can never increase variance (though it need not be as efficient as generating an independent set of random numbers).
\begin{aligned}\operatorname{Var}[(X+Y) / 2]&=\frac{1}{4}[\operatorname{Var}(X)+\operatorname{Var}(Y)+2 \operatorname{Cov}(X, Y)]\\&=\frac{\operatorname{Var}(X)+\operatorname{Cov}(X, Y)}{2}\end{aligned}
Now it is always true that
\frac{\operatorname{Cov}(V, W)}{\sqrt{\operatorname{Var}(V) \operatorname{Var}(W)}} \leqslant 1and so when X and Y have the same distribution Cov(X,Y) ≤ Var(X).
Related Answered Questions
Question: 11.EX.23
Verified Answer:
Let m(t)=\int_{0}^{t} \lambda(s)\, d s[/lat...
Question: 11.EX.16
Verified Answer:
(a) They can be simulated in the same sequential f...
Question: 11.EX.6
Verified Answer:
Let
\begin{aligned}& c(\lambda)=\max _{...
Question: 11.EX.3
Verified Answer:
If a random sample of size n is chosen from a set ...
Question: 11.EX.1
Verified Answer:
(a) Let U be a random number. If \sum_{j=1}...