Chapter 10: Q. 10.8 (page 431)

Suppose it is relatively easy to simulate from Fi for each i = 1, ... , n. How can we simulate from
(a) $F (x) = \prod_{i = 1}^{n} F_{i} (x) ?$
(b) $F (x) = 1 - \prod_{i = 1}^{n} [1 - F_{i} (x)] ?$

Short Answer

Expert verified

(a) The CDF is the CDF of maximum.

(b) The CDF is the CDF of minimum.

Step by step solution

Part (a) Step 1: Given Information

We have to prove the statement

$F (x) = \prod_{i = 1}^{n} F_{i} (x)$

Part (a) Step 2: Simplify

Suppose that follows the distribution with CDF $F_{i}, i = 1, \dots, n$ .
(a)
Observe that $F$ is the CDF of the $max (X_{1}, \dots, X_{n})$ . Indeed

localid="1648207466886" $\begin{array}{l} F (x) = P (m (X_{1}, \dots, X_{n}) \leq x) = P (X_{1} \leq x, \dots, X_{n} \leq x) \\ = \prod_{i = 1}^{n} P (X_{i} \leq x) = \prod_{i = 1}^{n} F_{i} (x) \end{array}$

So, generating $X_{1}, . . ., X_{n}$ considering its maximum, call it X. From the fact shown above, we have $X$ that has required CDF.

Part (b) Step 1: Given Information

We have to prove the statement

$F (x) = 1 - \prod_{i = 1}^{n} [1 - F_{i} (x)]$

Part (b) Step 2: Simplify

(b)

Observe that $F$ is the CDF of the $min (X_{1}, \dots, X_{n})$ . Indeed

$1 - F (x) = P (m (X_{1}, \dots, X_{n}) > x) = P (X_{1} > x, \dots, X_{n} > x) = \prod_{i = 1}^{n} P (X_{i} > x) = \prod_{i = 1}^{n} (1 - F_{i} (x))$

which implies

localid="1648207223842" $F (x) = 1 - \prod_{i = 1}^{n} (1 - F_{i} (x))$

So, generating $X_{1}, . . . . ., X_{n}$ and consider its minimum, call it $X$ . From the fact shown above, we have $X$ has required CDF.

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Suppose it is relatively easy to simulate from Fi for each i = 1, ... , n. How can we simulate from
(a) $F (x) = \prod_{i = 1}^{n} F_{i} (x) ?$
(b) $F (x) = 1 - \prod_{i = 1}^{n} [1 - F_{i} (x)] ?$

Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Simplify

Part (b) Step 1: Given Information

Part (b) Step 2: Simplify

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Suppose it is relatively easy to simulate from Fi for each i = 1, ... , n. How can we simulate from(a) F(x)=∏i=1nFi(x)?(b)F(x)=1-∏i=1n1-Fi(x)?

Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Simplify

Part (b) Step 1: Given Information

Part (b) Step 2: Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Suppose it is relatively easy to simulate from Fi for each i = 1, ... , n. How can we simulate from
(a) $F (x) = \prod_{i = 1}^{n} F_{i} (x) ?$
(b) $F (x) = 1 - \prod_{i = 1}^{n} [1 - F_{i} (x)] ?$