Q. 10.9 Suppose we have a method for sim... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 10: Q. 10.9 (page 431)

Suppose we have a method for simulating random variables from the distributions F1 and F2. Explain how to simulate from the distribution
F(x) = pF1(x) + (1 鈭� p)F2(x) 0 < p < 1
Give a method for simulating from
$F (x) \{\begin{cases} \frac{1}{3} (1 - e^{- 3 x}) + \frac{2}{3} x & 0 < x 鈮� 1 \\ \frac{1}{3} (1 - e^{- 3 x}) + \frac{2}{3} & x > 1 \end{cases}$

Short Answer

Expert verified

Observing flipping a coin. If, it falls Heads, declare $X = X_{1}$ otherwise, declare $X = X_{2}$ .

Step by step solution

Given Information

Consider

$F (x) = p F_{1} (x) + (1 - p) F_{2} (x) 0 < p < 1$ and

localid="1651486678603" $F (x) = \{\begin{cases} \frac{1}{3} (1 - e^{- 3 x}) + \frac{2}{3} x & 0 < x 鈮� 1 \\ \frac{1}{3} (1 - e^{- 3 x}) + \frac{2}{3} & x > 1 \end{cases}$

Simplify

Let us consider $X_{1} ~ F_{i,} i = 1, 2$ which we can generate. Now, generating random number in $(0, 1)$ , call it $U$ . If $U < p,$ declare $X = X_{1}$ . If $U 鈮� p,$ declare $X = X_{2}$ . Because of the fact that $X$ assumes value $X_{1}$ with probability $p$ and value $x_{2}$ with probability $1 - p,$ we have that $X$ has CDF

$F (x) = p F_{1} (x) + (1 - p) F_{2} (x)$

To solve the second part of this problem

$F_{1} (x) = \frac{1}{3} (1 - e^{- 3 x}) + \frac{2}{3} x F_{2} (x) = \frac{1}{3} (1 - e^{- 3 x}) + \frac{2}{3}$

Now, we will write

$F (x) = 蠂_{(0 < x 鈮� 1)} (x) F_{1} (x) + 蠂_{(1 < x)} (x) F_{2} (x)$

Considering random variable $G ~ E x p o (1)$ and define $P (G 鈮� 1) .$ Now, we generate random variable X.

$F (x) = p F_{1} (x) + (1 - p) F_{2} (x)$

and using the first part of this exercise, we have $X$ has distribution.

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91影视

Suppose we have a method for simulating random variables from the distributions F1 and F2. Explain how to simulate from the distributionF(x) = pF1(x) + (1 鈭� p)F2(x) 0 < p < 1Give a method for simulating fromF(x)13(1-e-3x)+23x0<x鈮�113(1-e-3x)+23x>1

Short Answer

Step by step solution

Given Information

Simplify

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

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