# How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $$\lambda$$ is $$3\lambda$$ ?

I've come to this conjecture from simulation with the following Java code

package stochastic.processes;

import java.util.function.DoubleSupplier;

import org.apache.commons.math3.distribution.ExponentialDistribution;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;

public class CoxProcess implements DoubleSupplier {

private final ExponentialDistribution intensityDistribution;

RandomGenerator generator;

public CoxProcess( double intensity )
{
this( new Well19937c(), intensity );
}

public CoxProcess( RandomGenerator generator, double intensity )
{
this.generator = generator;
intensityDistribution = new ExponentialDistribution( 1.0 / intensity);
}

@Override
public double getAsDouble()
{
ExponentialDistribution dist = new ExponentialDistribution(generator, intensityDistribution.sample());
return dist.sample();
}
}