Chapter 6: Q. 6.27 (page 273)

If $X 1 / X 2$ are independent exponential random variables with respective parameters $λ 1$ and $λ 2$ , 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 $f_{z} (z) = \frac{λ_{1} λ_{2}}{{(λ_{1} z + λ_{2})}^{2}}$

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

Step by step solution

Given information

X₁ and X₂ are independent exponential random variables.

Parameters $λ_{1} and λ_{2}$

The joint density function is

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

Explanation

The joint density function:

F_{Z} (z) = P (Z < z) = P (\frac{X_{1}}{X_{2}} < z) = P (X_{1} < z X_{2}) = \int_{0}^{\infty} \int_{0}^{2 y} λ_{1} e^{- λ x} λ_{2} e^{- i y} d x d y = \int_{0}^{\infty} (1 - e^{- 4 z}) λ_{2} e^{- λ y} d y = 1 - \frac{λ_{2}}{λ_{1} z + λ_{2}} = \frac{λ_{1} z + λ_{2} - λ_{2}}{λ_{1} z + λ_{2}} = \frac{λ_{1} z}{λ_{1} z + λ_{2}}

The distribution of Z is

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

hence the distribution of z is

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

Compute P (X₁ < X₂)

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

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

Short Answer

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Explanation

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If X1/X2are independent exponential random variables with respective parameters λ1and λ2, find the distribution of Z=X1/X2. Also compute PX1<X2.

Short Answer

Step by step solution

Given information

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Geometry

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

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