Question

Suppose that X₁ and X₂ form a random sample of twoobserved values from the exponential distribution with parameter \({\bf{\beta }}\). Show that \(\frac{{{X_1}}}{{\left( {{X_1} + {X_2}} \right)}}\) has the uniform distribution on the interval [0, 1].

Short answer

The proof has been established.

Step-by-step solution

Given information

Suppose X₁ and X₂ form a random sample of two observed values from the exponential distribution with parameter\(\beta \),

\({X_i} \sim \exp \left( \beta \right),i = 1,2\)

Standard theorems on exponential and gamma distribution

Need to use the following results to accomplish the proof.

If,

\(\begin{array}{c}\,{X_i} \sim \exp \left( \beta \right)\\{X_i} \sim Gamma\left( {1,\beta } \right) \ldots \left( 1 \right)\\\sum\limits_{i = 1}^r {{X_i}} \sim Gamma\left( {r,\beta } \right) \ldots \left( 2 \right)\end{array}\)

Standard theorems on gamma distribution and beta distribution

Now,

\(\begin{array}{l}If,\,X \sim G\left( {\alpha ,\beta } \right)\,\,and\,\,Y \sim G\left( {\lambda ,\beta } \right)\\ \Rightarrow \frac{X}{Y} \sim Beta\left( {\beta ,\beta } \right) \ldots \left( 3 \right)\end{array}\)

Also,\(c \times Gamma\left( {\alpha ,\beta } \right) = Gamma\left( {\alpha ,\frac{\beta }{c}} \right) \ldots \left( 4 \right)\)

Using the theorems

Therefore, by using (2)

\(Y = {X_1} + {X_2} = Gamma\left( {2,\beta } \right) \ldots \left( 5 \right)\)

Therefore, by using (1) and (4)

\(\begin{array}{l}{X_1} \sim Gamma\left( {1,\beta } \right)\\\beta {X_1} \sim Gamma\left( {1,1} \right)\end{array}\)

By using (4) and (5)

\(\begin{array}{l}Y = Gamma\left( {2,\beta } \right)\\\beta Y = Gamma\left( {2,1} \right)\end{array}\)

Finally,

By using (3)

\(\frac{{\beta {X_1}}}{{\beta Y}} \sim Beta\left( {1,1} \right)\)

Finding the pdf

\(\begin{aligned}{}Beta\left( {1,1} \right)&= \int\limits_0^1 {{x^{1 - 1}}{{\left( {1 - x} \right)}^{1 - 1}}dx} \\ &= \int\limits_0^1 {1dx} \\ &= 1\end{aligned}\)

This is the pdf of Uniform distribution [0,1].

Hence,

\(\frac{{{X_1}}}{{{X_1} + {X_2}}} \sim U\left[ {0,1} \right]\)

\(\)