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A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 6: Q. 6.27 (page 273)

If $X 1 / X 2$ are independent exponential random variables with respective parameters $位 1$ and $位 2$ , find the distribution of $Z = X 1 / X 2$ . Also compute $P \{X 1 < X 2\}$ .

Short Answer

Expert verified

The distribution of Z is $f_{z} (z) = \frac{位_{1} 位_{2}}{{(位_{1} z + 位_{2})}^{2}}$

$P (X_{1} < X_{2}) = \frac{位_{1}}{位_{1} + 位_{2}}$

Step by step solution

01

Given information

X₁ and X₂ are independent exponential random variables.

Parameters $位_{1} and 位_{2}$

The joint density function is

$f_{x_{1} x_{2}} (x, y) = 位_{1} e^{- 4 x} 位_{2} e^{- 位_{2} x}$ for x, y $> 0$

02

Explanation

The joint density function:

F_{Z} (z) = P (Z < z) = P (\frac{X_{1}}{X_{2}} < z) = P (X_{1} < z X_{2}) = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{2 y} 位_{1} e^{- 位 x} 位_{2} e^{- i y} d x d y = {鈭�}_{0}^{鈭�} (1 - e^{- 4 z}) 位_{2} e^{- 位 y} d y = 1 - \frac{位_{2}}{位_{1} z + 位_{2}} = \frac{位_{1} z + 位_{2} - 位_{2}}{位_{1} z + 位_{2}} = \frac{位_{1} z}{位_{1} z + 位_{2}}

The distribution of Z is

$f_{2} (z) = \frac{d}{d z} F_{z} (z) = \frac{d}{d z} \frac{位_{1} z}{位_{1} z + 位_{2}} = \frac{位_{1} 位_{2}}{{(位_{1} z + 位_{2})}^{2}}$

hence the distribution of z is

$f_{z} (z) = \frac{位_{1} 位_{2}}{{(位_{1} z + 位_{2})}^{2}}$

Compute P (X₁ < X₂)

$P (X_{1} < X_{2}) = P (\frac{X_{1}}{X_{2}} < 1) = P (Z < 1) = \frac{位_{1}}{位_{1} + 位_{2}}$

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