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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();
    }
}

enter image description here

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