Given a random variable $X$ folloiwng truncated exponential distribution on interval $[0,1]$ with parameter $\lambda$, i.e., $$f(x)= \frac{\lambda}{1-e^{-\lambda}}\cdot e^{-\lambda x}.$$ It is direct to derive its expectation $\beta =E[X] = \frac{1}{\lambda}- \frac{1}{e^\lambda-1}$.
Now given $\beta_1$,$\beta_2$, $a_1$, and $a_2$, satisfying $a_1+a_2=1$, and $\beta_1+\beta_2=\beta$.
My problem is: Can we find two independent random variables $X_1$, $X_2$ distributed separately on interval $[0,a_1]$ and $[0,a_2]$, and satisfying $E[X_1]=\beta_1$, and $E[X_2]=\beta_2$, such that
$$X=X_1+X_2?$$